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I bought a bike on Ebay which has a Sora groupset. The rear derailleur is noisy and jumps in and out of gears despite much tweaking. I've given up on it.

What derailleur can I buy that will work as a drop-in replace of the Sora derailleur without my having to also change the cassette or shifters?

The Sora is either medium or long cage - not sure which. It has approx 3.25 inches between jockey wheel centers.

Cassette has 32 teeth on largest ring.

EDIT: Just found someone else with the same problem. Mine sounds much the same as this: Noisy rear derailleur (Shimano RD-5800)

EDIT2: Motivated by your comments and after a bit of a clean and a bit more of a systematic re-indexing as per the video linked in the accepted answer (which I just changed below) the derailleur noise is in fact much improved. It's still not perfect but it's good enough.

TY!

As a sort of addendum to Nathan Knutson's answer, focusing on things to do before replacing the derailleur.

1. Make sure wheel is fully in the dropouts and parallel to the frame.




2. Properly and clean the drivetrain with a degreaser and apply a bike chain specific lubricant, not penetrating oil such as WD-40, GT85 etc. There are many videos that show how to do this.




3. Check hanger alignment. From the photo it looks pretty good (cage should be parallel to the chainrings). It might be worth getting a bike store to check it with a hanger realignment tool.




4. Replace cables and housings. Stretchy cables and sticky housings can cause the derailleur to not index properly.




5. Run through a systematic derailleur adjustment process.




6. Check chain and cassette wear.

For 9-speed (or 10-speed, which uses the same actuation and usually works fine in this instance) with native 32t clearance, the options for a Shimano road group model are pretty limited due to how the chronology played out with 32t road cassette being a thing. A lot of the 11-speed road GS cage derailers can clear 32, but they won't work with your shifter. The one you have and Claris RD-2400-GS may be the only nominally road ones that would work as a drop in replacement. RD-2400 is okay, barely.

Another answer is to get any GS or SGS (much more common) mountain 9-speed derailleur, which are compatible because Shimano 9-speed mountain and road rear derailleurs use the same actuation ratio, and all the 9-speed mountain RDs clear a 32. Of these there are a lot of choices; some currently produced ones are RD-M590, RD-M591, RD-M4000, and RD-M772. The XT RD-M772 may be the only one currently produced that comes in mid-cage/GS like you presumably need. That would be a good choice for a bike that's going to get used a lot.

Of note is that it's exceedingly common for rear derailleur hanger alignment as well as cable issues to cause the problems you're talking about, and you really shouldn't go to replacing the derailleur until those are eliminated as possibilities, unless you can tell the RD itself is obviously bent or slopped out, which does happen. And if the RD itself is bent, usually the hanger will be too from whatever incident(s) caused it, which will need addressing for a new RD to work correctly.