I email my T.. average twice a week, I think. When I want/need to vent, when I've come to a insight or when I feel I didn't explain something well enough in session, and when I think of something important I know I'll never dare bring up in session.
She encourages me to call 24/7 (literally! Nights, days off, weekends, everything) when I'm feeling particularly bad, anxious, have urges to hurt myself (even when they're "easy" to ignore) or have suicidal intent. I never do.
I actually dislike it when my T responds with a long email. Therapy is for during sessions; email is to give me a place to "park" my thoughts.

I email my pdoc for med issues. Sometimes he responds by calling me.