I am adding this summary following what was discussed in Juntemos en respuestas wiki las respuestas cortas específicas de regiones / Let's use community wiki to summarize set of short region specific answers. Feel free to edit to add the term used in your country or region.
Argentina
- Sobaco is much more informal, almost vulgar. You'd never hear it in a deodorant ad, or in medical speak.
Bolivia
- Sobaco is really vulgar. Better use axila.
Chile
- Sobaco is also almost vulgar and very informal.
Colombia
- Both forms are used about equally.
Dominican Republic
- Sobaco has the same meaning and connotations that in other Spanish speaking countries. Axila is the more ethical and formal, while sobaco is more informal.
Ecuador
- Sobaco is more informal, I think it is better to say "axila".
Mexico
- Sobaco is vulgar and if you do use it, it is when someone has bad odor in their armpits (te huele el sobaco, hueles a sobaco). Axila is when you are actually referring to the armpit as a body part.
Panama
- Axila is the word that we use to refer to the armpit. Sobaco is used for people from the "Guetto".
Peru
- Sobaco is not very used, the usual form is axila for both formal and informal situations.
Spain
- Sobaco is more informal than axila.
Cuba
In Cuba we use both Axila and Sobaco. Sobaco is more informal than Axila. But there is another way to say it, even more informal than Sobaco. I would say that is really really vulgar: grajo.
Grajo is used when someone has bad odor in their armpits (tienes olor a grajo, o tienes peste a grajo). This is the vulgar way used in Cuba.
