Superman is in two books a month, and therefore has a larger than normal rogues gallery but no Framing tokens. The Superman comic focuses on his life in Metropolis, while Action comics covers his adventures in time, space and around the globe.
"Superman" Comic villains:
Bruno Mannheim and Intergang: A persistent and growing problem in Metropolis, backed by advanced technologies of unknown (to Superman, at any rate) origin. Readers are aware that Brainiac has managed to insert technology into the brilliantly benevolent mind of Professor Emily Hamilton, technology which manifests as a separate villainous personality: The Atomic Skull. This alter-ego hides its existence from Professor Hamilton, and has created a secret alien weapons facility in a hyperspace pocket from which she anonymously supplies Mannheim and others as part of some as-yet-unrevealed Brainiac master plan. The Atomic Skull has not yet directly confronted the Man of Steel.
Mxyzptlk: the imp has twice been banished by Superman's cleverness, but it's only a matter of time before the enigmatic entity returns...
Minutia: A product of intergang's technology who has since moved off on her own, minutia can shrink things. Anything. A lot. On one level this lets her have a variety of high tech tools and weapons at dust size on her body so that she can unlimber almost any gadget she needs, but that pales before her ability to really screw with things. There are plenty of plots where whole scientific complexes are missing because she shrunk them and took them. Needing to escape Superman she lobs out a school bus full of kids that had gone missing earlier that day. An entire district of metropolis is held hostage under a drop of water. Superman has to fight his way out of an anthill, or is trapped between the molecules of an electromagnet.
Recluse: This spider-themed villain has a grudge against the daily planet, or superman, or both. His four artificial spider arms are kind of a poor man's doc ock, but the top two fire darts or clouds or sprays of varying toxicities of spider venom (that might even be enough to hurt superman if he ingest it or gets it into his bloodstream) and the bottom two fire a webbing like glue too strong for Superman to break without extreme effort. He's the villain who shows up and inadvertently threatens Clark's secret ID, because everyone in the Planet officers gets hit why isn't he effected? Or he and Lois get glued their hands together during the initial assault and he has to resolve the situation without giving up the secret ID. Or Recluse lays his hands on one grain of Kryptonite and glues it to Superman.
Remus: the requisite Magic villain, Remus is the ultimate cursed werewolf. Fast, strong and tough enough for his magical claws to hurt Superman, he appeared once every full moon to hunt Metropolis. After their first two encounters Superman managed to discern his blameless secret identity, Hunter Moon, and work out a scientific cure that seemed to work, but now Hunter turns into Remus on just the 13th full moon of the year ten times more powerful than he was before, able to slash Superman's flesh and Hunter has gone into hiding, the magical curse masking him from Super-senses. If Clark is going to find him again it will take real investigative reporting. Like Mxyzptlk he is now an occasional problem, but a very real one.
Chimera: an early and now permanently defeated Superman villain. A criminal mastermind who worked independently from Intergang, Jackie Reynolds, ran 'the worst gang of Suicide Slum', and was renowned as being 'the toughest guy' in Metropolis until Superman showed up. He was always legally protected from his crimes, but the need to best Superman overwhelmed him and he partook of with ever more bizarre mutation devices and serums giving him an array of one-off powers (kinda like Jimmy Olson) until in Superman #40 he takes them all at once has a massive blowout battle with Superman that ends with Reynold's heart giving out.
Lex Luthor, philanthropist, billionaire family man, and technological genius, was mayor of Metropolis and a key ally of the superheroes during the 2012 crisis. His wife and daughter died as part of the massive collateral damage during one of Superman's battles of that fateful year. He (and his equally brilliant teenage son) are consumed by anger and resentment over this. Superman is becoming aware of Lex's vengeful machinations (when he determined that the Metallo cyborg had previously been Luthor's brother-in-law and head of security), he has no legally actionable evidence, and is somewhat conflicted about how best to expose/deal with this threat without causing further strain to Lex's son and the still-recovering Metropolis community.
Action Comic Villains:
Brainiac: This hyper-intellect is dispersed across a network of almost unfathomably advanced AI technology. Its interactions with (mostly) humanoid sentients began as a learning subroutine, but Brainiac's serial manifestations in green humanoid form have led to an "addiction" for a single humanoid perspective and emotional personality. Superman has met several of these Brainiac avatars, and each has had a slightly different appearance, agenda, and demeanor (although all have been manipulative of humanoid life and disdainful of anything that might be considered morality).
Bizarro: Pushing the extreme limits of his speed to return to Earth from a distant corner of space, Superman found himself in a distorted, somewhat nightmarish reflection of his adopted planet, complete with a similarly mad version of himself as its champion. This Bizarro being was belligerent and irrational, if not exactly evil, and his following of Superman back to "reality" nearly ended in interplanetary disaster as he attempted to "fix" things in terms of his otherworldly sensibilities. He is banished and his odd world sealed off for now, but his raging vows to return and "save" Superman and this galaxy have the Man of Steel convinced that it is only a matter of time before Bizarro finds a way to return.
Mongul/War World: Superman's encounter with an expansionist, militaristic, interplanetary empire led to his capture and gladiatorial enslavement, and only his defeating of the Emperor Mongul in single combat was enough to buy a decree that Earth and its neighboring systems would be spared. How long this decree will be honored is uncertain, and the Man of Steel has countered several "unsanctioned" actions by Mongul's "rogue commanders."
Cryana Cordave: The Ice Crow is the queen of her race and committed to making the last of the Kryptonians her bride (and assimilating his DNA into her childline. Self-centered, very powerful and unwilling to take no for an answer she spotted Superman after his Warwold incident and has used her super-powered royal guard, teleportation technology, and other things to force the Kryptonian into her presence so that she can pull him more permanently into her orbit. Her family line has incredibly potent ice generation powers, and while they claimed the throne through use of that power Superman has to admit that her world is beneficently ruled, meaning that the tactics that world on Warworld won’t work here. (This is a sideways homage to Lilandra and the Imperial Guard, though Cryana is much more personally powerful and her Royal Guard is less powerful and versatile than the Legionesque Imperial Guard. Still, they have about a dozen members, any of which can show up to cause Superman grief.
Wang Mang: the man who would, for a while, be an emperor of China foresaw the forces that would lead to the defeat of his planned cultural retrenchment – especially after an encounter with the young Man of Steel – and developed a long term plan to reach his ultimate goals. A very long term plan. As in, his plan will not see fruition until 2030. But the tools of this ultimate conspiracy have woven themselves into every corner of human society, even as one splinter group of it, the cult of Kobra, makes all the flash and noise. Superman will occasionally – usually when travelling through time but not exclusively so – encounter some centuries in the making plan or trap, and will also jump back to the beginning of the common era to directly confront Wang, whose powers rest entirely on persuasion and plotting.