My DD9 rides many rides alone.
Goofy's Barnstormer, Indy Cars, Tea Cups, Magic Carpet & Dumbo, and Big Thunder Mountain (5 times in a row on 7/1 and 9 times in a row on 7/3 - she got to be great friends with the CMS!) at MK. She has also done Primevil Whirl & Triceratop Twirl at AK. With a FP I think she would do Test Track alone.
Depending on the wait time, some times I go in line with her other times, I wait near the exit. We are local and spend a great deal of time at the parks, so she is very comfortable in her surroundings and the "process". BTMR is really the only of these rides that if I do not get in line with her that she is out of my sight for most of the wait time. These are rides that I don't care for or that make me feel ill, and this is they only way she gets to ride since she is an only child and I am a single Mom. I feel comfortable with this bit of independence for my DDs maturity level. I know she respects the rules (she is a rule follower by nature) and at this point would not try to do anything that might be dangerous or get her in trouble on the ride!!
She will not do Space Mtn alone - she loves the ride, but does not want to be in the dark without someone she knows, she also likes RNRC and TOT, but has not asked to go on them alone.
BTW I was told at Primevil Whirl that if she was 7 and met the height requirement she could ride alone - she was 8 and met the requirement at the time.
I know this is a decision that each parent needs to make for their child/ren, but it works well for us!