It depends more on your preference than the type of bow. If the child has fine hair, I tend to go with an alligator clip. French barrettes are used mostly with older girls since they have more hair to hold that barrette. Also, I don’t use the non-slip grips. If you line your clip correctly you don’t typically need that, but it can’t hurt if you want to use it as a extra selling point and if your customer is requesting it. I’ve found that the extra bulk in the clip actually makes it more difficult to work with in your girls hair and doesn’t always stay better. I prefer to properly line my clips with grosgrain ribbon and haven’t had an issue with slipping. Now the French barrettes will tend to slip more easily since they are typically larger than alligator clips and also because the way they are designed, the hold isn’t quite as tight.
The only bow that I would say is pretty much mandatory to use a specific type of clip would be a French barrette for a loopy bow. Although I have seen bow makers create small loopy bows on alligator clips, so it’s not out of the question.