Equalizers, by nature, shift phase. However, it's rarely an issue because under normal conditions the effect is simply not audible.
The truth is that we cannot hear phase shift. We can only hear the side-effects of phase shift when we add two versions of the same (or very similar) signal that are not in phase with one another.
For example, if you record one source with two microphones, it's possible that by using different (and extreme) EQ settings for each microphone you could conceivably hear comb filtering at some frequencies. More likely, just the fact that you used two microphones is going to be a far greater concern. Presumably, the OP is singing into just one microphone.
So-called "linear-phase" equalizers shift phase, too. They just shift all frequencies by the same amount so they remain in sync with one another across the spectrum. It's one of the reasons linear-phase EQs are primarily used on the master bus, because the delay doesn't matter when filtering the entire mix. A linear-phase equalizer would still cause comb filtering under the conditions described above, except that it might be even more noticeable because it'd be across the board, not limited to a particular frequency range.
Extremely narrow notches (and boosts) can cause another problem - not related to phase shift - called ringing. This is what happens when the Q is so narrow that the filter is on the edge of being an oscillator. It's usually something to be avoided, but it applies to all equalizers, uh, equally. You can't avoid it by choosing a different EQ.