How is a "good" or "bad" singer defined in signal processing terms?
First order, some DSP analysis might be able to detect central pitch, pitch variation (vibrato), timbre (overtone richness), and note onset, and say whether these are grossly out-of-tune, thin, or off-time, by how much and how often.

However, second-order, good musicians play with subtile variations in pitch (equal vs. just intonation, barbershop quartet harmonization as the key changes, etc.) and in timing (jazz, blues, "swing" rhythms, etc.).  How to "score" the latter is probably more a topic for computer AI than DSP.