First, some cartridge manufacturers will void their warranty if liquid cleaning is used. Be aware of the provisions of YOUR cartridge's warranty before you do it.
Nandric's post above was spot on with regard to one risk. If a stylus is attached with glue, and if we don't know the composition of said glue, no one can say whether any particular liquid solvent is safe or a disaster in the works. Additionally, the elastomers used in some cartridge suspensions may be weakened by volatile organic compounds present in some liquid cleaners.
Elizabeth's blithe spirit notwithstanding, I'm personally aware of three instances where liquid cleaners apparently weakened the stylus/cantilever bond, leading to a lost stylus.
I disagree with Nandric's final statement that, "In many cases there is no substitute for wet cleaning. Look at your stylus with a magnifying glass to see for yourself." I have, with dozens of cartridges. A stylus that's properly maintained with the Magic Eraser method is as clean as new. Liquid cleaning will not make it any cleaner.
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As you've seen, many people use and get good results from the Magic Eraser method I first publicized ten years ago (wow! time flies). I still do.
The "how-to" questions are all addressed in the original post linked by jmcgrogan2 above.
How often to clean? JCarr (Lyra cartridge designer) educated me on this soon after that post. His statement was that, even if an LP is perfectly clean, stray vinyl particles are loosened from the sidewalls with every play. Heat and pressure at the stylus/groove interface can bond these to the stylus. To prevent a gradual buildup that discolors the stylus and (worse) smothers the sound, JCarr recommended cleaning the stylus AFTER EVERY SIDE.
The ME, used as suggested, safely removes/prevents this buildup. I've been using it for 11 years on cartridges that retail up to $10K with nothing but positive results.
I've also rehabilitated a fair number of supposedly "worn out" cartridges sent to me for inspection, simply by cleaning their styli properly. In only one case did the ME fail to remove all the built up gunk. That one was so covered in burnished-on gunk that I resorted to a strip of ultrafine sandpaper, as Linn used to provide with their cartridges. In all other cases, and certainly with regular use, the ME and a brush or goo dip to remove stray particles is all that's required.