As for their abilities, Supes has the power but very different code of ethics from Bats. And Batman is by all account a muc much smarter character than Superman. One-on-one fight with no enhancement and Superman would pulverize Batman... Bats has no superpowers. On the other hand, as shown in Dark Knight Returns, with some Kryptonite brass knuckles and a lot of pre-planning and Batman could exploit easily every little weakness that Supes has.
Should they fight? No. They are both heroes afterall.
