This is a pretty cool topic, especially considering I don't know exactly what I consider Shin Godzilla at this point.
I guess I will break it down by appearance, and then behavior.
Dorsal plates, usually 3 rows (varying degrees of size and shapes). A typical look for these would be a "maple leaf" look, but can be a bit more complex.
Has a heat beam, and prior to use the dorsal fins will illuminate. I prefer blue, but at this point the color has varied quite a bit.
Within the above construct, I'd like to see a design that doesn't overly deviate from tradition too much. This means that Godzilla has had 3 or 4 toes, he doesn't need 7. Godzilla has always had 4 fingers - don't give him 12. Godzilla should maintain the classic Godzilla silhouette we are all accustomed to, but I would expect several tweaks to his appearance from eye color, dorsal fin rows, size, etc., but nothing insane like giving him 2 heads or wings.
Has the typical Godzilla roar
Impervious to conventional weapons
Does not run from the military
Yes, it's true that over a period of 60 years, Godzilla has gained or showcased a few new "powers." With the exception of the nuclear pulse and red beam from the Heisei era, the rest of the "abilities" were really one offs from movies made in the 70's for films that appealed to kids and at a time when Japanese cinema was in the toilet. I really consider things like the tail slide, Godzilla flying, and his magneto power to be the exceptions rather than the rule. He doesn't use those outside the specific movie they appeared in and they happen ONCE in the franchise.