I recently used "Ultra Soft Auto Wenol Polish" on a brush handle, and the results were stellar. Really, really good. Straight up AMAZING. It's a very fine grit polish, so it won't buff out scratches that are very deep. It also takes a bit of elbow grease to really work it in, but if I hadn't seen the results with my own eyes, I wouldn't have believed it. More importantly, it removes very little material, which, in my opinion, is paramount. Brasso might put a nice sheen on your handle, but when you're done, you'll only have half a handle. :-D
Purchase with care, though, because Wenol makes different formulations. You want the one in the light blue tube, which seems to be the least common variety. I doubt you'll find it locally, as it seems like somewhat of a specialty product. It looks like it was designed to polish thin chrome plating without removing enough metal to get through the chrome. It says "non-abrasive," but that's just not true. Nor would you want it to be true, really. Good polishing requires a little abrasion. Otherwise, you're just cleaning. But it will do amazing things to a lightly scuffed handle, and did no damage to the Rooney brush I used it on. I even inspected it with a jeweler's loupe when I was finished. Flawless. FWIW, it sells on Amazon for $10.99, free shipping.
It really depends on the scratches and material. I am a big fan of Meguires PlastiX and find it works great. Flitz has always been by go to metal polish but it does work on plastics with some success. If the scratches are deep enough you may need to use some sandpaper to get them out. In that case I would suggest 3k wet/dry or even 4k, dipped in water with a not of dish soap. Then after you have sanded it smooth polish to a mirror finish. Be careful sanding some things as it can change the color unexpectedly. Some plastics fade over time to often pleasing colors and all to frequently people start sanding them only to find that great butterscotch color gone. Post some pics of the ones you are planning to polish up!
Good advice above. If you're unable to find Flitz locally, Maas performs just as well in my experience, although I haven't had much luck finding either Flitz or Maas locally. I also am a bit fan of PlastX. These all are excellent products. Good luck!
Meguires plastic cleaner wil remove scratches. I also have used Simichrome for decades for polishing razors, kinives, stainless Smith & Wessons, cell phone screens and shaving brushes. When you get black on your rag, you have a very mild abrasive that brings out a great shine on plastic. It's made in Germany and available online, AG Russell, motorcycle shops etc. It comes in a tube.