Superman has super-hearing. It's obviously not literally sound that he hears, since he can hear something at the instant it happens despite being on the other side of the world (Adventures of Superman #632) or even on another planet (Infinite Crisis #5), but he is more or less omniscient in this respect. With this in mind, when a lot of people are in trouble (anywhere in the world, like Belize or under the ocean or something), why don't they all shout out "Superman!" at once to get his attention?
Superman is a Kryptonian. His homeworld, Krypton, orbits a red star. Superman gains almost godlike powers under yellow sunlight, but red sunlight takes those powers away. What's the deal there? Why - how - would a species evolve to take advantage of a type of energy that it was never exposed to during its evolution, but be harmed by its natural environment?
Kryptonite is shown to harm Superman from a distance due to radiation. Presumably this is radioactive decay, so what does kryptonite decay into?
Come to think of it, how come the only mineral that can hurt Superman is found exclusively in the form of remains of his home planet? How come no kryptonite is shown to occur naturally anywhere else in the DC universe? Doesn't that seem like a bit of a coincidence?
In Grant Morrison's "Rock Of Ages" storyline in JLA #10-15, a dystopian alternate future was shown in which Darkseid ruled the Earth. To prevent this future occuring, it was necessary to prevent Superman from destroying a near-omnipotent artifact called the Worlogog (or Philosopher's Stone) in the present - presumably so that it can be used to stop Darkseid at some point in the future. When, exactly, is this going to happen? (My guess, based on increasing recent New Gods activity: soon.)
Does Batman ever stop and wonder how he has managed to survive literally thousands of fights with bad guys with guns? The odds in a given fight are pretty good - let's say 99.99%, but over the course of all his adventures, those odds dip into serious danger territory. Does it never occur to him that he should be dead at least once by now?
Come to think of it, consider Animal Man - who, as of Animal Man #26 (1990), is vaguely aware of the fictional nature of his existence. If he became fully aware of the fact that he is a long-running and well-loved character, and that because of this it is exceedingly unlikely that he will ever be killed off, could he use that fact to his advantage? Could he communicate that fact to Superman and Batman, who, by the same token, are actually immortal?
What happened to Luke McDunnagh, Plastic Man's son in JLA #65? That was a great story, and it was followed up pretty well in "Trial By Fire" (JLA #84-89). He is a great character and has potential. But now it seems Plastic Man's son is an unremarkable black-and-white fellow named Ernie (52 Week 43 et al) - first seen in The Kingdom in 1999. I'm pretty sure this is intentional and they're going somewhere with this (The Kingdom hasn't been specified as out-of-continuity, and see Justice Society Of America #2), but the question is still out there.
More to come if I think of them. I just thought those were worth throwing out there.