A good way to find the right singer for you is to listen to their samples. Their samples are likely a good representation of the style, color and quality of their voice. Read reviews from previous clients, and make sure to always contact the singers
Hiring a session singer is typically done with one of the following arrangements:
Background Singers will sing backgrounds to your existing vocals. For example, they will double your vocals, sing harmonies or ooh and ahhs.
Demo Singers will sing the lead vocal part for your song for the purpose of shopping your song to labels or artists. The recording with the demo singer's vocals isn't intended for commercial release, since it's not 'featured'. Demo vocals are intended to showcase your song in the most impressive way possible - with a killer vocal performance. This is important for getting the interest of labels, publishers, or the artist you'd like to record your song. The vocal performance is prominent in how people listen to music and judge songs. If you don’t have the vocal melody and lyrics aren't already written when you approach a singer, you might need a topliner or writers to write the part, if the singer you are hiring is a performer more than a writer. Many topliners also sing, so there’s some crossover between singers, topliners and songwriters.
Featured Singers will sing the lead vocal part for your song or track for commercial release. Typically, featured singers will charge more for the privilege of releasing the track commercially with their vocals, and depending on what was agreed, using their name and/or likeness as part of the release.
Singers who also write - Many singers are songwriters. If the song is not completely written and the singer writes all or some of the vocal melody or lyrics (as opposed to singing a part that was already written), they will typically expect to be recognized as a co-writer, unless you specifically agree the writing is 'work for hire' (aka a full buyout) which may cost more. It is typical for co-writers to keep a share of the 'publishing' - the royalties associated with songwriting, unless a buyout arrangement was agreed to.
Topliners or 'top line writers' are writers who help write the vocal melody (and sometimes lyrics), to a finished backing track. This is common in EDM, Hip Hop and EDM/R&B influenced Pop. Any genre where the 'track' or beat was written first, and the vocal melody and lyrics after. Producers and DJs bring a finished track to the topline writer to write a vocal line to the track. As long as the backing track is essentially fixed and finished, and someone else is asked to write the vocal melody (and in some cases lyrics), they are referred to as topliners. Many topliners are also demo singers - they can deliver a great sounding vocal performance of the 'topline' they wrote. Make sure to clarify if the vocals being delivered by the topliner are demo vocals (to showcase and shop the song), or featured (released commercially with their vocal recording).