Part of the answer on how to clean these things involves knowing right away what dirt and stains are and aren't removable and part of it is taking a chance. I've thrown away or passed on my fair share of irreparably stained or broken clothes, pillows, electronics, everything. Here are a couple of things I often find looking super filthy that clean up rather easily.
Wedgwood Jasperware loves water. Don't be afraid to clean it. It's also not very porous so most stains do come out with a minor amount of elbow grease. If the stains look stubborn, I start by giving it a good soak in some soapy water for a couple of minutes.
For any stains that don't come right off after a soak, I wet a soft bristled toothbrush, pump some soap onto it, and gently rub into the detail spots where stains and dirt tend to persist. Then I'm left with this:
Dirt is almost entirely gone from the bottom (the rims are sometimes difficult because they are a little more porous than the rest of the piece) and is only left in a few of the more intricate parts of the design on the front. This is an amount of clean I can live with for something that was fully caked in stains and left to sit sometime between 1962 and now. I periodically attack stains like this again when I'm cleaning something else. I might try a toothpick (or anything sharp but soft) the second time around to get a little more of the stain off.
Avoid: Any stains where you can see a grease line extending out past the stain. That has soaked in forever and ever.
See that line of what looks a little too rusty to be simply called oxidation on the inside? It clearly goes through the plating? You have to know that stains like that are just never ever going to come out. If they're on the front/outside of an object, maybe don't buy it. This one is on the inside and is going to be camouflaged by the things I intend it to hold so no big deal to me.
For larger items, I like this kind of polish but this is a small piece and for that I like the cloth:
I feel like the cloth is less abrasive and it also cuts out the drying step. I wouldn't use the cloth for anything large because it takes a long time but I think it's worth it to keep small, detailed items nicer longer. I do a general but thorough surface rub, then hold the cloth over a cotton swab or orange stick to get into the fine details, then do a final allover buff. For the details, you want something as close to the size of each crevice as possible to get it cleaned quickly.
Doesn't it look so happy to be my new brush holder?
So, here I don't want to do the step to replace the cloth on the bottom right now (this thing is missing parts, after all) so I can't just dunk it in water like I would normally do with something this intricate and dirty/dusty. In cases like these, baby wipes are my best friend.
I get the first levels of surface grime off with the wipe. Then, because I have a mirror surface, I take some glass cleaner (doesn't hurt the metal if you use a "natural" one).
I never spray directly on the object. I sprayed on a paper towel (the less linty the better here) for the mirrored surface, then attacked the edges of the mirror and all the intricacies of the metal with a cotton swab sprayed with the glass cleaner.
My paper towel was a little bit linty so I waited for this to fully dry and then rubbed the surface with a piece of a microfiber towel I cut up into smaller pieces. Microfiber is also my best friend. It's not a very exclusive club.
This will hold my lipsticks, as it was originally intended to do. Hope this helps everyone who has asked me questions. I'll keep doing this kind of post as I acquire more things that need cleaning.