Xbox music on your phone will play mp3 files you add to your phone.
What you have to consider is that Xbox Music is a lot of things, there is a web based streaming service (Free up till the new year), a service for Xbox music pass holders, an app on PC's, Xbox's and phones, if you take the trouble to understand it, Xbox music is mostly trouble free (It is for me).
Because of the way they all potentially work together you do need to think things through.
For the first few months of my usage of the Xbox music app on my phone all I used it for was mp3's and it works, got to admit after looking at all the options I ended up taking out a music pass and now have a collection of over 3000 songs, more than half of them from the music pass and the whole thing works very well, but I match my local songs across devices by actually copying them across.
I don't know what PC's you have, but in Windows 8 8.1 there is a setting you need to approach with caution and that is the one called "Automatically match my local music to my cloud collection".
If this is turned on it will match local music with the Xbox music service and then allow any local music you have to be played on other devices, but it is potentially a real nuisance as some songs are not available because of copyright issues for example, the Beatles and Led Zeppelin do not allow their catalogue to be used so if you turn on the above setting you will end up with a very small collection, 8 Led Zeppelin albums end up as one track.
No need to sign up to Xbox music to just play your local songs as Ven07 said, just copy your mp3's across and it should work, but if you already have clicked the "Auto" setting on any of your PC's, you may need to do a bit of sorting out.
Bob