Godzilla and Superman are both very iconic characters; known for their power and nigh invincibility. However while having them fight sounds like an entertaining fight; the fact remains that Superman dominates this battle due to the superior magnitude of his abilities.
===Speed===
The biggest issue Godzilla has no answer for is speed. Godzilla is known for being a lumbering giant; he is a tank even compared to other kaiju. The most agile characters Godzilla have faced (like Megaguirus and Rodan) are at best about hypersonic in their speed. Yet Superman has been confirmed to fly faster than light on multiple occasions, and has both raced and 'tagged' the Flash on multiple occasions: confirming that Superman is on par with him in speed. Even if we severely lowball Superman; the fact remains that Godzilla is so relatively massive and slow that he has no chance of dodging any of Superman's attacks; if Godzilla can even react at all.
===Strength===
Strength is another trait that both of these characters are well known for. Godzilla has proven to be able to KO other kaiju with just a few blows and has thrown kaiju much larger than himself. One of these feats include throwing Kumonga (30,000 ton kaiju spider) over the horizon (which is about 12 miles away). As impressive as it is throwing a kaiju across the Earth; Superman is strong enough move the Earth ITSELF. While it is difficult to confirm the limits of both characters in terms of strength; the magnitude of strength is clearly different. If Godzilla tried to attack California again; Superman could literally lift California away. Godzilla can destroy buildings, but no film has depicted him this strong.
===Defenses===
Both of these characters are considered indestructible; but this word has been quite overused, even for characters that technically are not indestructible. Godzilla is able to survive nuclear blasts at point blank range; while The Dark Knight Rises depicts Superman as being almost killed in such a blast. However it should be noted that Superman is weak to nuclear radiation. This is part of the reason Atomic Skull, Nuclearman from Superman IV (best movie ever) and kryptonite can harm Superman. It should also be noted that Superman's weaknesses and limits are rather inconsistent. In the Justice League episode The Doomsday Sanction; this nuclear missile has a kryptonite warhead, as Cadmus saw it necessary for they concluded that Superman could survive the blast.
But since I mentioned I would analyze Superman by the comics; I will show the comic examples are even more concrete to my argument. Red Son Superman survived a point-blank explosion of '6 miniature black holes' with a '15 million mile blast-radius'. Superman has also survived supernovas or attacks on par with supernovas on multiple occasions. Burning Godzilla meanwhile died when he self-destructed; an explosion that had the potential of destroying Earth (if the military's freezing weapons didn't cool it down). Godzilla has also been harmed by Gigan's blades, Hedorah's poison and from basic attacks from other kaiju; none of which can destroy a solar system like a supernova can.
Now the fact that Godzilla uses radioactive attacks arguably means that he can harm Superman. But with Superman's aforementioned speed advantage; it would be difficult to even hit him to begin with. It's also safe to say that Godzilla doesn't have the strength to harm Superman enough to guarantee him a victory.
===Verdict===
I honestly do not see how Godzilla could win this battle. Superman has too many advantages; every trait about Superman overshadows Godzilla. There is nothing stopping Superman from throwing Godzilla into deep space, or into the phantom zone, or just One-Punch-Man-ing him with an Infinite Mass Punch. I don't see Godzilla producing enough radiation to harm Superman, or Godzilla finding any way to counter Superman's speed advantage: which could end the fight in less than 1 second.
Godzilla was designed to be an unstoppable threat to mankind. He did quickly evolve into being designed for kaiju-fighting; but with so many kaiju being giant mutant dinosaurs, Godzilla was constantly facing threats that confirmed his mortality. Godzilla fought conventionally, primarily against opponents that had some slight advantage like superior size (Ghidorah / Biollante), numbers (Kamacuras), speed (Rodan / Megaguirus), durability (Mechagodzilla 2), etc... Yet the majority of the time, it was a fair fight with either monster getting a chance to win.
Superman meanwhile rarely faces a physically superior opponent; since being SUPERman literally means being SUPERior. If you look at Superman's rouges gallery; almost none of them are conventional. He has multiple evil clones (Zod, Bizzaro, Cyborg Superman), kryptonite users (Warsuit Lex, Metallo, KRYPTONITE MAN), and just strait up overpowered abilities that anyone would be screwed over by; like reality warping (Mxypitalik), adaptation (Doomsday) and power-stealing (Parasite).
At least with Space Godzilla; his crystals are weakspots and Godzilla had MOGERA as an ally. At least with Destroyah; he's not as powerful as the Oxygen Destroyer itself (that would make that duel end much faster if that was the case) and Godzilla was in 'Burning Mode'. At least with Megaguirus, she doesn't turn Godzilla into a floppy potato after draining him, it's not as broken.
Godzilla faces physical challenges. Superman faces gimmicks. Godzilla can be overpowered. Superman IS overpowered.
===Side Notes===
We actually have seen Superman defeat Kaiju multiple times. The short 'Artic Giant' potentially influenced the creation of Godzilla. Even then; Superman dominated the kaiju in terms of strength. I'm not saying this Kaiju is exactly the same as Godzilla; but it is noteworthy that Superman had no difficulty fighting monsters like this (unless Lois Lane is there to be completely braindead).
