I have the exact problem. 2014 Model S. RWD
When I back out of the driveway in the morning I get a clunk from the left front tire area. Then when I go forward I get a clunk again on the first stop. I do sometimes get it at a stop after having not braked for a while (using regen instead) but it ony happens at slow speed as I'm coming to a stop. I can brake at high speed and never here it. The brakes feel fine then but coming to a stop I hear the clunk and feel it through the brake.
I replaced the sway bar connector as it had a cracked seal - didn't fix it
I removed the brake pads, cleaned them up and lubricated them - didn't fix it (new pads at 160,000km. 190,000km on it now so they are barely worn)
Control arms look good and are solid.
By happenstance I spun the rotor by hand while my friend pressed the brake and I got a clunk. I can now recreate the sound on demand by spinning the rotor one way and braking and then spinning it the other way and braking. I've uploaded a video (sorry about the pump noise as it was hot in the garage and the cooling kicked on)
One thing I have noticed is the brake caliper "flexing" when the brake is applied. Is this normal? I have not pulled the other tire yet to check the other side. I've uploaded a second video showing the caliper flexing out as the brake is applied.
Any thoughts on this anyone? Kinda stumped. Thinking replacing the brake caliper might be the solution.