Godzilla:
Giant lizard mutated by radiations.
Class 100 Strenght
Superhuman resistance to injuries
Superhuman reflexes and reaction time
Superhuman speed
Superhuman leaping abilities
Superhuman healing abilities
Nuclear Breath
Sonic Breath, showed when he roar and the shockwave send the cars flying, not burning them.
This are the powers of the Godzilla in Rol Emmerich's movie.
He was killed from missiles fired by F-18 airplanes, so his skin is not unbreakable, adamantium claws on the hands of a Class 800 Pounds guy like Wolvie can cut through his skin not with ease but can.
But onestly
Wolverine
Class 800 Pounds Strenght
Peak Human reflexes, reaction time and speed
Peak Human resistance to injuries
Superhuman senses
Superhuman stamina and Healing Factor
Adamantium Skeleton, and Claws.
Godzilla, in both Japan and U.S.A. versions, is a huge and tall monster.He can simply stomp on Logan, or make skyscrapers and buildings falling on his head, Logan would be severally injured and will fall unconscious.This is a battle he can't win.
You can pull out that Logan fought with Hulk, but Hulk cannot stomp him so easy like Godzilla, not to say Hulk is weaker, but just to say, Godzilla is giant, so it would be easy for him use the city and his dimensions against an enemy with the dimensions of Logan.It's just like Galactus fought with Logan, he can simply stomp his body.
The only way Logan is gonna killing Godzilla, is with Godzilla sleeping, Cyclops blasting into Godzilla's body at full force, and Logan being throwed to the heart of the monster by Colossus, passing through the heart.But that's a dumb situation, so the truth is Logan lose and gets wiped out.