Captain America vs. Batman is dicey. If the fight lasts a long time, Steve beats Bruce because of his endurance allowing to him outlast Batman, who would eventually tire out. And while Cap is the better fighter (due to his experience and physical advantages; Batman is the better martial artist), Batman is willing to fight dirty. If Batman chose to attack using gadgets, and not going mano y mano, Cap wouldn't stand a chance. Because Batman would have something that would negate that as an advantage or would simply get around it. 

The fight I'd like to see? Batman vs. Iron Fist. No utility belt or iron fist, so neither has an extra advantage over the other.

Does Wolverine’s healing factor repair the kind of nerve strikes that a skilled martial artist like Batman or Shang Chi might employ?  Because thats the only way I can see anyone who isn’t superhuman (in strength or ability) could damage him.

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So I'll assume we're talking about A) they're fighting for some reason, and B) they're battling all out.

What skills do they possess? All are in perfect physical shape*. All are familiar with at least a few fighting skills. Most have special enhancements, and Batman has his brain and his utility belt.

*By nature of the super soldier serum, Captain America is always in perfect physical shape.

I'll preface this by noting that I feel Mr. Byrne was entirely correct that Batman and Captain America (or Bruce and Steve) might battle evenly for a few minutes - until they realized who they were fighting, and agree to stop. But this seems to relegate this to just "Let's you and me fight!"

Daredevil's martial abilities are hand to hand, gang style, and perfectly suited against more normal villains.

Iron Fist studied the martial arts of K'un L'un, which I think are superior to Daredevil's skills.

Batman has learned 'most every martial art, and has the utility belt, nightsight lenses (among so many other gimmicks), and his brain. Not that any of the others are dummies... but Batman's only power is his intelligence, and he uses it to best effect.

Captain America is a combat veteran, against Nazis, Hydra, Kree, Skrull, and so many others - and I am pretty sure he learns from everyone he battles. He knows more combat styles than anyone. He is just the ultimate battle man... even more so than Batman.

So my choice is Cap - again in a total out and out fight with no other considerations.

There are a lot of other fighters - Robin, Nightwing, Black Canary, Wildcat, Hawkman, Hawkgirl, Karate Kid, Shang-Chi, Hawkeye**, the Shroud, Nighthawk, Moon Knight, the Falcon, Black Widow, the Winter Soldier, and so many more. I think Batman or Captain America beat any of them***.

**Yeah, I know that Hawkeye has his bow, Hawkman, Hawkgirl, Nighthawk and Falcon can fly, and so on. I believe we're putting those aside for this circumstance. 

***Batman vs. Robin? Cap vs the Falcon: Geez...

And then Wolverine. Mr. Byrne was right... all a combatant needs to do is get him angry, and it becomes all instinct. Not enough to outsmart Batman or Cap.

I think brains and experience combine to show the best fighting ability... well, and a friendly writer!

Nobody we are talking about here could have survived that punch from Jahf.  Everyone mentioned would be a wet smear on the little guy’s fist.

Wolverine has survived fights with the Hulk. 

Cap shouldn’t be able to deck Wolverine with a punch in the gut.  At least not in the book I’d write.

Captain America and Batman are experts at all forms of combat (as is, incidentally, Karate Kid from the Legion of Super-Heroes.) So surely I'd pick from one of them.

I think Captain America as the pre-eminent fighter tends to get overblown. I think in 1940, his training would have been the best given to the best of the soldiers which would have lasted weeks. He then spent roughly 4 years fighting soldiers, the odd super-nazi, but few really elite trained fighters. Since, his un-freezing (the amount of time ago depending on the vargaries of Marvel Time, but not decades), he had done far more training than he did beforehand and fought more super-villains. Although, his training is rarely against other highly skilled hand-to-hand combatants, he's often the one in the sensei role and guiding the training of others, raising them to be better. And, it was consistently shown that skilled un-augmented opponents can be his equal. I think it is generally forgotten that he won his first bout with Batroc partly because Batroc was past his prime. A few years younger and the fight might have gone the other way. Since that first encounter, Batroc's age is rarely brought up and is treated as simply one highly skilled opponent vs. another.

On the other hand, Iron Fist has trained with premier martial artists since childhood to become the best of the best.

DD through the Silver-age also had accelerated healing and not strictly a normal human, but until he went all ninja in his background training, he was a skilled fighter whose edge was chiefly his expanded senses.

Unpacking Batman is a bit more difficult because of all the layers of retcons added to his character over the years. Like Captain America, I tend to think his pure fighting skill has been elevated a bit too much over the past years. But, like the others, looking at the core character concepts as created and shown in stories before a lot of the power drift came in -- like Iron Fist, he has trained from a young age. Like the others, he has a personal mission and/or tragedies that drive him to excel. However, Batman's training is more diffused and fractured than others because he's training to be a criminologist and detective. He must not be merely a good fighter, but excel at various sciences and forensics, psychology, criminology. He studies weapons, locks, safes, etc., Unlike the others, there is nothing beyond his drive and natural gifts to give him an edge in a straight up fight against someone not intimidated by his reputation and look. Thus, a middle-aged Ted Grant Wildcat is able to go toe-to-toe with a Batman in his prime, and a street gang with martial arts skills might actually be able to get the drop on him temporarily.

Part of what dictates the outcome would be the situation. In a ring with combatants in fighting trunks and no distractions and other elements that can be used for improvisation, it would come down to Captain America and Iron Fist. Iron Fist is the best pure fighter of the lot, training in almost exclusively in that for years. Captain America's super-soldier serum body offsets that with optimum conditioning, strength, and reflexes as well as a background in fighting with more different styles. It would come down to which would get the crippling punch in first.

A different situation where they were fighting in a room, a rooftop or a back alley could swing the edge away from Iron Fist back towards Batman and DD where Batman's more varied training and fighting styles and DD's heightened senses to allow him to be hyper aware and make use of a more chaotic environment are able to come into play and it could be up for grabs just depending on specifics and who lands the first punch.