There's always going to be a problem about asking "how common" something is in speech. It depends on a person's dialect and the area where they live, it depends on the situation, and it depends on personality. Both forms, cited by the OP, are common though.
Speaker A: Hello, what's your name?
- Marilou
- It's Marilou
- Hi, I'm Marilou.
- My name's Marilou
The phrase My name is Marilou has a quite formal tone, the only time I've used it was in presentations, or when I had to teach about introducing yourself in English.
This American English video, which was made by a teacher called Rachel, is really good. It also explains which syllables should be stressed in the phrase. my NAME's MArilou
Some non-native speakers don’t want to use contractions because they don’t think it’s clear enough, but we really do want to use the contraction “I’m”, and not “I am” because it can be much quicker, I’m, which puts the emphasis on the name, the most important part. This will also help smooth out your speech.