Does the Logitech F710 work on the PS3?

I can confirm F710 works on PS3, just make sure the switch is set in the D (legacy) position. No vibration feedback though :).

I can't speak to the F510 or F710 specifically, but the PS3 does support HID devices (computer USB game controllers). I was just able to successfully play a few minutes of Ratchet and Clank Future: Quest for Booty on my RumblePad 2 (an older gamepad from the same series).

There are two caveats. First, rumble was completely non-functional — either the PS3 doesn't support HID force feedback or the how rumble works has changed since my device was made. Anecdotal evidence suggests that the DualShock 3 is itself a HID device, however, which is promising.

Second, I can't find any information on whether the "Nano-receiver" used by the F710 works with the PS3. I'd recommend either going with the F510 (which is corded, so has no receiver) or get your gamepad from a store with a good return policy.