It is becoming more popular. I just looked at and rode a horse today that the owner is not quite ready to sell yet. She loves her and has owned her for 5 years. My riding instructor is also her riding instructor, and he thinks that the mare and I would be a good fit. I am not quite ready to buy yet, and this way the mare will be moved to my RI barn, I will get to learn on her, take care of her, the owner will know she is in a good place, and hopefully after a while, I'll be able to buy her.
The owner will also be able to come up and visit and/or ride if she likes.
Also, something to keep in mind. At least in my area, for everything I would have to do to have a horse on my property, plus purchase price of the horse, tack, etc. it is less expensive to start out this way. I am so new to horses after many years away, that it makes sense for me to lease for a while.