To me, 部 is a formal classifier for 手机. 台 is usually used for a bigger machine, like computer & TV 一台电脑 & 一台电视. 一台手机 doesn't sound quite right to me.
一把手机 is not used in practice for whatever reason. I guess it might be because 我要买一把手机 could mean you want to buy a handful of 手机s, implying several 手机s, not just one.
Colloquially, sometimes I heard 一个手机 is being used. Like in the sentences: 我要买一个手机; 我想买个手机.
Well, 我要买一部手机 sounds more formal and it's not as casual as 我要买一个手机 or 我想买个手机 in our daily communications.
That's my observation from my neck of woods. Others from other regions may take this differently.