I've read that some billing software offer your clients possibility to register domains along with registering for hosting services.
What software does it, and to what domain registrars should I register to use these features?
It supports registrations via eNom, OnlineNIC, OpenSRS & SRS Plus. What registrar you choose to use is something that only you can answer after doing a bit of research. I personally think eNom offer an excellent service - it depends upon your needs.