Is it correct to say “在手机”？
Your sample sentence should be "我在我的手机上听音乐" as ChineseHulu.com said.And you actually SHOULD not omit the word "上" in this sentence,or it would be a little bit weird to native Chinese people. 

"在...上" is a preposition phrase.It can be used on representational target like:
在操场上 -- on playground
在沙发上 -- on sofa

And it can be used on abstract target like
在手机上 -- on cell phone
在电脑上 -- on computer
在网络上 -- on internet

Usually we Chinese won't omit "上" if we use this phrase on abstract target or express specific position relation.So these sentence should not omit this word:
我正在手机上玩游戏 -- I'm playing game on cell phone.
我想在电脑上看电影 -- I wanna watch movie on computer.
我通常在网络上聊天 -- I usually chat on internet.
男孩在沙发上跳着 -- The boy is jumping on the sofa.

But if we use this phrase to express the location where something happened,not focusing on the position relation.We can omit this word:
男孩们在操场上玩 or 男孩们在操场玩 are all ok.(Boys are playing on the playground.)

And if you wanna express something happened on the specific location but it's a large area.You MUST omit the word "上" and the phrase thus turn into another single word preposition phrase "在...".
我在上海工作 -- I'm working in Shanghai.
我在苏州旅游 -- I'm traveling in SuZhou.

Apologize for my English.Hope you will get it all... :D