I think I have seen it spelled both ways.
Is one spelling more common or formal than the other ? (e.g. in the U.S.)
According to the New Oxford American Dictionary (that I have by default on my Mac :)):
The spellings adviser and advisor are both correct. Adviser is more common, but advisor is also widely used, especially in North America. Adviser may be seen as less formal, while advisor often suggests an official position.
Since it's an official position, I'd rather go for Ph.D. advisor
I grew up learning the spelling as "advisor," which goes along with "supervisor." I've never seen "superviser," either, and "adviser" just looks strange to me.
This may actually be a field-dependent issue: in academia, I've always seen "advisor" as the preferred spelling (and a number of schools agree with that assessment). However, outside of academic contexts, "adviser" seems to be preferred, both in the UK and the US, as shown here.
In my high school, my teacher said it is British English Vs American English.