It depends, is the answer. And it also depends on what you call "distortion" - do you mean it in the sense that a guitarist would, or just that the sound is changed?
Microphones are the first potential source of distortion. Sometimes you want a "smooth" mic, but sometimes you want one which puts a bit more "grit" inio the sound. Mics are fairly consistent as they come from the manufacturer, but mics from different manufacturers and of different construction will produce a noticeably different sound.
After that, you have the mic preamp. On solid-state electronics, turning the gain up so that the signal clips (reaches the maximum voltage possible) causes hard edged distortion which can sound pretty nasty. I've used it as an impromptu effect in the past though for someone jacking an electric guitar in directly, but it's not normal.
On valve-based electronics though, for starters they distort everything. Your basic distortion level is higher than on solid-state. However that distortion is initially mostly in even harmonics. As you push the gain up, a valve preamp doesn't go straight into saturation but instead just progressively distorts more, adding more odd harmonics as it gets crunchier. This is more commonly referred to as "overdrive". It's still distortion, but it's under a degree of control, and subjectively it sounds appealing to us.
Mostly a sound engineer would try to keep mic preamps out of this region, and stay within the most linear range of the preamp. The Rolling Stones though famously pushed all their mic inputs into overdrive, because they wanted that sound. Some other recordings used it more by accident, or couldn't get away from it because their gear was inherently not that good (for example, "Louie Louie" by the Kingsmen).
Muse use distortion on vocals as an effect - check their cover of "Feeling Good", amongst others. It's an effect though, not a key part of Matt Bellamy's normal vocal sound.
You certainly can add some in, and if you think it sounds good then fine. But ultimately though it all does have to start with a good voice.