I think Mayson's labeling Superman as a vigilante had less to do with his powers and more with his anonymity. Can there be due process when the person making the citizen's arrest is anonymous?
Superman isn't anonymous. He's Superman from the planet Krypton. It's not the same situation as, say, Batman. Batman is obviously someone else who is hiding his identity and could reasonably be considered anonymous. Superman is not known to have a secret identity, so there's no reason to believe that he's someone unknown hiding his identity.
In Silver Age comics, it was commonly known that he had a secret identity, but he considered Clark Kent to be a civilian cover for his true identity as Superman, so I think the public still wouldn't have considered him to be anonymous back then, either.
How the courts would rule on using your own senses if they are super would be a unknown. They do allow things like drug sniffing dogs that have better then human senses but are natural.
I would say that Superman would be under the same restrictions that a normal human would have, i.e., something would have to be in plain sight for someone with normal sight. Your point about drug-sniffing dogs shoots a hole in my idea, though. On the other hand, Superman has way more enhanced senses than any dog, to the point that anything not hidden by lead and encased in a sound-proof (not just mostly-sound-proof) room would be "in plain sight". The room would also have to be air-tight to thwart his super sense of smell. That does not seem reasonable and would likely lead to unlawful searches under the reasoning that, "It's in plain sight for Superman, but he wasn't around today to help us, so we did ourselves with technology and searches what he can do with just his senses." It seems like a slippery slope.
Then there's the question of his microscopic vision. Can he zoom in to analyze microscopic residue on the countertop if in the same situation a standard cop wouldn't be able to take samples and send it to a lab? Or are cops allowed to swab anything they want as long as it's in plain sight?
All in all, I'm not sure I like the idea of Superman operating as an officer of the government. In the '60's Batman TV show, Batman and Robin were deputized members of the police force, but that worked because they were just cops with special gadgets and unusual means. They were subject to the same limitations as the other cops. (I don't think that modern Batman would work quite as well as a cop.) Superman is on a whole different level. When he starts taking orders from the police chief or the mayor, governor, or the president, things start to get scary. Politics gets involved, and it's a whole Pandora's box. I think he works much better as an independent good-Samaritan type, given that he has the moral compass that we expect from Superman.
Given that, he should endeavor to work with law enforcement insofar as he should not destroy evidence when possible and apprehend crooks and turn them over to the police in a way that enables them to be properly prosecuted. I don't think that would really get Mayson off his back, but it would weaken her case against him.
