If you have a convertible hi-back booster, you will be allowed to use it on board only if the 5-pt. harness is in place; if the harness has been removed, it is not allowed.
Seat-only boosters (aka "belt-positioning boosters") are not useable on US-based air carriers, but some European airlines do allow them to be used. BTW, someone mentioned the Baby B'Air; that is not a tandem harness; you cannot use it to restrain a child on your lap. It can be used to restrain a child who is sitting in a seat by himself, but it cannot be used during takeoff and landing, for some reason. Also, if you have a small baby and use a sling or snugli-type carrier, don't be surprised if the FA insists that the child be taken out of the carrier for takeoff and landing.
We just had a nasty turbulence incident near our airport here. The plane had to make an emergency landing because about half of the passengers on the flight needed medical attention. Remember, if the aircraft hits turbulence and drops suddenly, an unrestrained child is going to fly head-first into the bottom of the overhead bin. I've been in bad turbulence incidents, where the only thing keeping me from hitting the ceiling is the seatbelt, and I'm no featherweight.