Removing stains from acrylic (porcelain-look) faucet handles

have you tried plastic cleaner?

https://www.whatsupfagans.com/how-to-clean-faucet-sink-handles-globes/

What sort of stains are they and what room is the sink in?? Try taking them off and soaking them in hot, soapy water with a few drops of bleach. Let them soak for a little bit. Check to see if some of the dirt is coming off by wiping with a sponge.

Maybe try baking soda and vinegar paste or a magic eraser.

If they are truly dirty and not yellowed (yellowed plastic will never be white again) , then I've had terrific results using a product called "Awesome" you can get it at most dollar stores

Thanks Charlee -I will try Awesome --I actually have some in my supplies right now --and yes, they are dirty -not yellowed --but the dirt is like a stain --really sunk in to the white fake porcelain.