What type of iPhone would be compatible with Assurance Wireless service?

So I've used an iPod touch for years, but now that my old assurance wireless free phone is no longer working right, I'm wanting to use some money gifted from a relative to basically put everything on my iPod touch onto an actual iPhone... but I'm worried about compatibility with assurance wireless when there's few options available through them.

I've seen some worrisome talk around the internet about incompatibility so if there's any suggestions on how I can get a quality phone that will be compatible with that network I would appreciate it.

[Re-Titled by Moderator]