Intellect
Already very smart by nature, exposure to the same chemicals that altered his appearance altered his mind. His intellect expanded into a very unique variety of genius. When he is in the right frame of mind to apply his skills, he does so with amazing prowess. When learning something new, he can go from completely lacking in understanding to mastery in a few short years.
Agility
What he lacks in the durability and brute force of his favored foe, he more than makes up for in agility. He is measurably faster and more agile. This allows him to perform extremely death-defying feats without losing his balance or missing his mark. It also adds to his fighting ability greatly when properly applied.
Healing Factor
So minor it has never been documented as even being there, Joker has a subtle version of healing factor. He doesn't heal any faster than anyone else, but what he does do is heal completely from things that would permanently cripple anyone else. Every bone in his body has been broken more than once, yet he has no real signs that any of them healed wrong and even when his mind has been forcibly reconfigured, it eventually returns to it's original state.
Chemistry
An incredibly skilled chemist, Joker is capable of concocting drugs and toxins undreamed of by others. This is most evidenced by the many variations of Joker Venom that he has managed to create.
Genetics
Certainly not the world's best geneticist, he is reasonably good at playing around with genetics when given the chance. Understanding what is going on with genetic codes or devices that manipulate them.
Engineer
While he very rarely creates very complicated devices himself, he often modifies the devices of others. This is highly evident when he steals any form of robotics or takes over abandoned amusement parks and the like. He shows a knack for weaponizing just about anything.
Nuclear Science
Good enough to poison a city or make a bomb, that is about as far as his skill extends in this area.
Electronics
Much like Engineering, his ability to fidget with the inner workings just as readily as the outer shells. While engineering skill tends to focus on death traps, he turns his electronics on sabotaging devices, subverting systems and fudging the controls of electronic devices.
Computers
Originally very weak in computers, he kidnapped more than one computer genius and has studied what they do. Even before he was capable of handling modern computers, he was already skilled at hacking into all sorts of communications devices. Even into the present, he seems to be able to get control over almost anyone's communication system unless it is military level encrypted. Sometimes even then.
Sleight of Hand
Sleight of hand is nothing special in most cases. He falls into the global magnitude only because of just how very few are able to take it to the extreme that he can. Short of being made of rubber and wormholes, most people can't manage to pull out tricks from thin air as well as this man. Even so, for those who are capable of doing so, he falls a little above average.
Hand to Hand
When he actually applies this skill, he has managed to go toe to toe with Batman. What he lacks in strength, he makes up for in dirty tricks and agility. Despite this, when it is time for him to lose, he seems more than willing to throw his own fight.
Explosives
A little more skillful in this area than with Nuclear tools, there isn't much to know about explosives that he doesn't already understand. Press the button, BOOM!!!
Master of Arms
He can be impressively accurate with a broad range of guns and blades. He employs any number of melee devices with effective skill and if you aren't a Superhero or Batman, you aren't probably going to get to avoid one of his aimed shots. With this said, he often comes across as incompetent because of his tendency to fire wild and let the person running decide which bullet hits by which direction they pick to run.
Jokes
While he can indeed tell a good quip, manage a great one liner or even make a twisted joke that gets someone to smirk, the Joker really isn't that good of a comedian. Between his generally bad sense of humor and his intimidating nature, very few people find any joke he makes all that funny.
Joker Venom
The effects of this toxin are nearly identical regardless of form. It's strength, delivery method and the rate at which it takes effect can be highly variable. One variety kills the target, one version causes permanent laughter until a counter toxin can be administered and one more creates a permanent smile to become fixed on the targets expression if not resolved quickly enough. The two most common forms are a breathable gas or a sprayable liquid. Another form is a powder that can be mixed with other substances. Only Joker is known to create and employ this toxin and he alone knows how it is created. He's even created versions that affect surprisingly powerful beings.
Quickshots
These are things like his trademark clattering teeth. The objects are tossed out and remain viable until destroyed or they find a target. Sometimes includes exploding objects or odd messages.
Lingering Toys
These are things like his trademark clattering teeth. The objects are tossed out and remain viable until destroyed or they find a target. Sometimes includes exploding objects or odd messages.
Heavy Hitters
Gag Guns that actually shoot after the gag, impossibly large mallets, and his joy buzzer of death all fall into this category. Bub and Lou, while Harley's pets, serve here. While not particularly complex, they pack a wallop and can severely injure even metahumans when employed properly.
Robots
More than once he has taken to using robots, either modified from someone else's design or unique creations of his own. These robots often have multiple functions, noteworthy strength and advanced AI systems.
Hypertech
While he very rarely creates his own hypertech, he regularly steals it. Once he has it, unless he finds its function particularly funny, he tends to modify these into strange new devices. The appearance of these devices in his bag of tricks is somewhat rare.
Mundane Weaponry
Explosives, Knives, brass knuckles, crow bars, tommy guns, machine guns and overly long handguns aside, Joker has access to one form of mundane weaponry that remains far less available to the common person. He gets ahold of high grade explosives as easily as one might pour water from the tap. While he has no shortage in the ability to create fancy bombs, he seems to favor more archaic forms of explosive. For situations not requiring a detonator, his item of choice is dynamite.
Pain Resistance
Several aspects factor into this, the most obvious being the changes in his body thanks to the chemical burns that bleached his skin. Most of his nerve endings are simply non-functional. Those that still function are given a boost to resist pain from the nature of Super Sanity (or madness depending on your perspective) making him have a very different view of things like pain. It is there, but he just doesn't acknowledge it unless it suits his mood and the nature of a conflict.
Poison Immunity
Years of chemical exposure and abuse, along with his altered biochemistry have led to a general immunity to almost every form of drug or poison. While this means the vast majority of toxins and poisons simply have no effect on him, of course, most of the positive drugs are of no use in helping him.
Madness Unending
It is possible with an effort to change the Joker's mind with magic, science or psychic manipulation. Changing how he thinks isn't the problem. Keeping it that way is. The chaotic mind of Joker refuses to stay sane for long and even the best effort will only last a minute or two at most. More often than not you only get a few seconds of change before the chaos of his mind returns to turmoil. For all intents and purposes, this acts like full immunity on a delay.
Run of Arkham
Not the most meaningful thing, but once he calls a place home, he somehow manages to gain abnormal access to all sorts of means of escape. Candy factories somehow end up having underground passages he finds that can let him get away. Fun houses turn out to be incredibly effective even before he modifies them. And of course, most notable is his presence in Arkham. Unlike other criminals who have to work hard at escapes, Joker seems able to just come and go as he pleases. No one can figure out how he escapes unless he goes out of his way to make a show of it. Sometimes, outsiders end up bringing him out, but sometimes he just seems to slip away. Even within the facility, he has shown an ability to get in and out of his cell without anyone seeing it happen and do things within the facility for his own twisted sense of humor.
Cheat Death
Over and over the Joker has fallen from cliffs, been blown up, been beaten to literal death. Each time he somehow manages to either evade inevitable doom or be brought back from his death. If this is an effect of his seeming knowledge of events he shouldn't know or simply the chaos of luck is unclear.
Atop the Fourth Wall
While he doesn't go so far as to utterly break the fourth wall in the way characters like Deadpool do, Joker is fully aware of the fourth wall most of the time in one degree or another. His knowledge goes beyond a single cannon. He can be aware of things from just about any source, even cross-overs or side media. He knows it is all just a show for the entertainment of someone else, but accepts his part in it. Oddly, the only person he ever really tries to bring into this same view is Batman. Even Harley isn't really aware of how he pulls information out of thin air. Batman though, he wants to see the truth. He wants batman's insanity to match his own for just one moment so that the one person who is so close to understanding him can understand him fully. Even then, he isn't willing to just explain it. One must go mad enough (super sanity) to see it on their own or no amount of explanation is going to work. This said, he is not above talking directly to his audience or referencing things that he should have zero knowledge of.
The Right Tool
An outcropping of his breaching the fourth wall, Joker seems to always have things set up just right in advance of a situation. Running away, seemingly at random, his timing can be incredibly perfected. Worse, he can land just right in the only safe spot and somehow already have a rope waiting for when he needs to swing away. Some unexpected element appears and there is a chance that he already happened to have just the right weapon sitting on hand for the situation. This is not every time though, since it is often an extension of the story. If plot dictates he gets away, he happens to have just the right things. If instead he is at the end of the 'story' he tends to only have what it takes to extend the fight in a climactic manner before going down.
Fearless
Joker has an absolute disregard for his own well being. If he lives or dies doesn't matter to him any more than the life or death of anyone else. It makes him both effective and ineffective at the same time.
Envenomed Blood
The same thing that makes him immune to poisons also has led to his blood having a permanent infusion of Joker Venom flowing through it. Blood to blood contact or Joker's blood in something ingested will lead to a non-lethal variety of the Joker Venom effect. While not deadly, it will incapacitate whoever has been exposed until an anti-venom can be administered.
Obsession
He refuses to actually kill Batman or even allow others to do so. So much of who he is is caught up in batman's existence that in the absence of batman he gets listless and just acts like a normal person to escape from his depression. Even when faced with nearly infinite power, he could not bring himself to completely destroy his foe.
Super Sanity
For all intents and purposes, this is insanity. Genius of the sort Joker possesses is so divergent that he breaks the fourth wall. He sees things not in terms of good or bad, but in terms of story. Cause and effect without judgment. A man shoots another man in the head while trying to rob him. We would see a killer and their victim. Joker sees a man pulling a trigger and a bullet going through the head of the other who fails to evade. Both are guilty, both are innocent. Events played out the way events were going to play out because that is how it had to be. The 'victim' was guilty of doing things that led them to that point. The 'killer' was drawn into this point in life by events they don't really have control over. Free will is an illusion and everything that happens is done for the sake of interesting story on behalf of some other world. It doesn't matter, it isn't real and even if it was, free will is an illusion anyway. One can regret killing someone, but there is very little real regret in erasing something from a piece of paper or deleting a bit of text on a screen.
Fearless
Joker has an absolute disregard for his own well being. If he lives or dies doesn't matter to him any more than the life or death of anyone else. It makes him both effective and ineffective at the same time.
One Man Show
He is utterly unwilling to share the stage with others. Most aren't foolish enough to try. His sense of humor sometimes fades when others try to be the jester around him. It has gone so far that he has very nearly killed his girlfriend Harley when she dared to try take the initiative on a 'gag'. While it does make him more dangerous, it also makes him sloppy.
Chaos
All of his history, indeed even his personality, are variable. Who he is today isn't who he was yesterday or who he will be tomorrow. He changes, adapts. He looks at the world and sees it for what it is. As such, he adapts everything about himself to match what he perceives in the moment. If the 'story' calls for him to get taken out in a single blow, so be it. If it calls for him to use every ability at his disposal and put his foe against the wall, so be it. Today he remembers being dumped in a vat of chemicals, tomorrow he remembers having elective surgery to look this way. You can never know who you are going to be facing or what they are capable of. He might be a harmless prankster now and a ruthless killer later.