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1.
Refers to a certain type of female cyclist in New York City. Hallmarks are long flowing hair, no helmet, wearing stylish footwear, a dress, jewelry. Bicycles range from expensive designer "city bikes" to dutch city bikes to vintage American and English 3 speeds, most equipped with baskets or luggage racks.

Etymology comes from BikeSnobNYC, the Godzilla part is presumably due to the fact that these women are wrecking New York City.
I was riding on 8th Ave, but had to jump out of the bike lane when a herd of beautiful godzillas were salmoning towards me with their shopping bags swinging all over the place.
by drag racer x April 15, 2010
 
2.
A woman who, when opposed by circumstances that would most likely result in failure for most people, is able to get what she wants with very little effort due to her charm and beauty.
"I got a speeding ticket but my beautiful Godzilla girlfriend got the cop to rip it up for me"

"I tried to break up with her but she's such a beautiful Godzilla that I wasn't able to do it any more"

"I'm late for work every day and never get in trouble. I love being a beautiful Godzilla"

"Even though the concert was sold out, my beautiful Godzilla friend some how got us front row seats"
by Charity Searcy July 27, 2011