We have always let the kids choose their activity, but only one at a time. They have to fulfill the season, term, session, or year before going on to something new. So my oldest did dance at 3 &4, switched to swimming at 5, then onto gymnastics at 6, onto ice skating at 8, and then Irish Dance at age 11 (which she is continuing with almost 15. My only exception to the ONE ACTIVITY rule is a musical instrument. So she has also been in the school orchestra playing viola since 4th grade. On and off, she has private lessons as well.
My 9yo did swimming at 3&4, and softball at 5. Then, began Irish Dance and is still in it. No signs of changing.
My 8yo boy did softball at 4&5, Irish Dance at 6 (was NOT his thing but he finished his year), and baseball at 7. Doing baseball again this year. As soon as he's old enough, he will do basketball in the fall and baseball in the spring. He's a sports FANATIC.
My 5yo did nothing till Irish Dance, which she just started. I was trying to make her wait till age 6 but she was positive she wanted to start. SHe'd watched the other dd's in their ID class enough to feel comfortable, I guess. She'll do her first competition in a couple of weeks.