— The styli, especially of the small and advanced types, tend to accumulate gunk as they play the discs. What can be done to clean it?
My personal choice is a polyurethane gel. There are many commercially available ones, from Audio Technica, DS Audio, ONZOW Zerodust etc. Gently lowering the stylus onto the gel does the job and has very little potential for mishaps.
There are dry brushes that are marketed specifically for the purpose. Some are manual, some have an "ultrasonic" actuator vibrating the bristles. "Ultrasonic" ones were very popular in the 1970s. I have tested a few vintage ones. While they do work in a way, they leave quite a lot of microscopic dirt behind.
Last but not least, there are combination brush/liquid systems, like the ones sold by Nagaoka, Audio Technica etc. These are also very effective. There are rumours of the cleaning fluids dissolving the glue that bonds the diamond to the cantilever — but neither me nor my friends have ever experienced anything like that.
— How do I check the tonearm bearings for slack?
— First, make sure that your tonearm is of the gimbal type, not a unipivot or a knife edge. Please consult the manual if you are not certain.
Doing this test will not harm unipivot or knife edge bearings, but you will probably observe some motion, often alarming in its amplitude. It's completely normal for these bearing types and has no relation to their health.
Vertical bearing: with the toneam on the rest (idle position), try gently rocking the central part of the tonearm from right to left and vice versa. Then try raising and lowering the central part, all without using force. A little bit of slack is ok, visible movement isn't.
Horizontal bearing: with the tonearm on the armrest and the headshell firmly attached, try gently rotating the armtube clockwise and counterclockwise as seen from the front. There should not be any slack at all.