Alright, I've got Post-Crisis Superman (because Post-Flashpoint is incomplete) and composite Godzilla up on VS Battles Wiki. Let's see how they compare, to see if one would just stomp the other and render this rivalry moot.
Tier: 4-B (Godzilla can get higher with Transformations, Power-Ups, Preparation, and Reactive Evolution), so that's a good start.
Godzilla Powers and Abilities (Base Form)
Superhuman Physical Characteristics, Hand-to-Hand skill, Large Size (Type 2), Self-Sustaining (Types 1 and 2), Enhanced Senses (including Sixth Sense), Instinctive Reaction, Danmaku, Immortality (Types 1, 3, and Limited 6), Regeneration (Low-Mid in battle, Low-High with some unspecified time, High over a significant amount of time), Stealth skill, extreme pain tolerance, Berserk Mode, Rage Power, Flight, Breath Attack, Energy Projection, Manipulation, and Absorption, Radiation Manipulation, Magnetism Manipulation, Heat Vision, Electricity Manipulation and Absorption, Fire Manipulation, Plasma Manipulation, Homing Attack, Accelerated Development, Adaptation, Reactive Evolution, Possession, Soul Manipulation, Forcefield Creation, Gravity Manipulation, Life Force Absorption, 4th Wall Awareness and Breaking the Fourth Wall, Vehicular Mastery, Anti-Spatial Manipulation, Limited Biological Manipulation, Duplication, Burrowing, Weather Manipulation
Intimidation Aura (specifically works against beings with enhanced senses), Telepathy (universal range), Paralysis Inducement, Electro-Magnetism, Vibration Manipulation (able to destroy things on a molecular level and negate durability)
Godzilla Resistances
Extreme heat (excess of 1000 degrees Celsius), mind control, acid, poison, electricity, fire, disease, possession, high sound waves, petrification, magic, matter manipulation, high gravity levels, black holes and absolute zero temperatures, sleep manipulation, limited resistance to Spatial Manipulation
Superman Powers and Abilities
Superhuman Physical Characteristics, Genius Intelligence, Stellar Radiation Absorption and Empowerment, Flight, Enhanced Senses, Information Analysis, Heat Manipulation and Limited Matter Manipulation via Heat Vision, Durability Negation via Heat Vision, Non-Physical Interaction (heat vision can counter intangibility and molecular phasing), limited Power Nullification (heat vision can cut off external power sources), Self-Sustaining (types 1 and 2), Super Breath, Ice Breath, limited Intangibility and Invisibility (via phasing), Afterimage Creation, Illusion Creation (via Torquasm Vo), Aura (protects clothes and avoids grime), Martial Arts and Pressure Point Strikes, limited Sound Manipulation, limited Air Manipulation, limited Electricity Manipulation, limited Vibration Manipulation, calming touch, Astral Projection, Regeneration (at least Mid-Low), Body Control (self only), Absorption of EM Energy, Geothermal Energy, and other types of Energy, can convert biological energy into Solar Radiation
Limited Space-Time Manipulation, skill with alien technology
Superman Resistances
Reality warping, Energy Manipulation, Soul Manipulation, Mind Manipulation, Fire Manipulation, Illusions, Empathic Manipulation, Radiation Manipulation, Matter Manipulation, Time Manipulation, Time Stop, Existence Erasure, Possession, Willpower Manipulation, Death Manipulation, Acid Manipulation, Electricity Manipulation, Magma Manipulation, Telekinesis, Deconstruction, Corruption, Biological Manipulation, Pain Manipulation
Attack Potency: Solar System level (Godzilla can go higher with Transformation, etc.)
Speed: Godzilla has Supersonic+ Movement Speed, Sub-Relativistic Attack Speed, FTL Reactions and Combat Speed, possibly Massively FTL+ Flight Speed. Superman moves Massively FTL+.
Lifting Strength: Godzilla is Class Y. Superman is Stellar.
Striking Strength: Solar System class.
Durability: Solar System level (Godzilla can go higher with Transformations, etc.)
Stamina: Godzilla's is Extremely high. Superman's is essentially infinite under yellow or blue sunlight.
Range: Godzilla's Atomic Breath is Planetary, and he also has interdimensional Teleportation. Superman's Heat vision is also Planetary.
Intelligence: Godzilla is gifted, Superman is a genius.
Godzilla weaknesses: Will explode if he absorbs too much radiation (he can easily recover from this). Destroying the second brain near his tail will paralyze him.
Superman weaknesses: Kryptonite, red stellar radiation, magic.