Assuming there's a finish on the fingerboard, naphtha (lighter fluid) works well. So will acetone if you don't have a lacquer finish. At a more mundane level, a damp soapy rag will do the trick too.
Am I seeing metallic lines? There's a future problem. Sure as long as you can keep from contaminating the pickups. That is why so many here recommend nylon scrubbers, no steel splinters to get concerned about.
I would think about working up to that. Start with a soft cloth probably. I don't think it's a good idea for the ongoing feel of your fingerboard to use anything that harsh.
I would never arrive at that. Ever. I have never heard of steel wool being used to clean finished furniture; a finished instrument is no different. The op finish is most likely nitro. That calls for naptha and or a mild detergent. Gentle work with a scotchbrite pad.
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