What Not To Use
Acetone and Acetone Nail Polish Remover
There's a good chance it will melt or discolor the plastic. Maybe not immediately, but over time. And is that a risk you really want to take? For makeup removal, sandpaper works well. If you don't want to use sandpaper on a non-matte faced doll, rubbing alcohol also works, but takes a lot of rubbing. By the way, acetone will also turn a shiny face matte, so you wouldn't even want that on a shiny faced doll to begin with.
Don't stick your pullips in the oven. OMG! OBVIOUSLY! right? Well, no, not obviously. For example, you are making something from sculpy that will mold onto a doll's head that if you take off and bake seperately will almost certainly lose it's shape. And you know that sculpy can bake at low temperatures like 175-200F, it just takes longer. That's nowhere near hot enough to melt a pullip head.
...but it is hot enough to shrink it. Pullip heads shrink in heat. Trust me. I have the shrunken heads to prove it.
Alcohol Markers (like sharpie, prismacolor, etc)
...will seep into the doll's skin and dye it. You will have to sand very deeply to remove it, if you can get it off at all. Markers in general = bad.
Oil-based anything (makeup, pastels, paints)
...will seep into the doll's skin and dye it. You will have to sand very deeply to remove it, and it may cause long-lasting damage to the plastic if you leave it on.
Does it clean up easily? Does it dry clear? Will it discolor, shrink, or otherwise change over time? Is it messy? Will the solvent necessary to remove it cause damage to the plastic? Will something else work better?
There is no approved/disapproved glue list by brand. When selecting a glue, these are the questions you want to ask yourself, and the sections below will explain why:
Generally, when at all possible, you don't want to use glue on anywhere the glue will be seen. Glue is inprecise, and tends to change color over time or become brittle and seperate from what it was originally glued to, crack, etc. Say for example you want to spread a thick coat of glitter over your dolls lips, or do a glitter body tattoo. Just smear on the glue, dust with glitter, blow away the remnants, or use a glue/glitter stick, just like art class in elementary school, right?
...No. In that case you would want to mix the glitter with a brush on gloss sealant or varnish approved for plastics and then apply the combination with a fine brush like paint to the area you want to coat. The varnish is safe for plastics, flat drying, and should (if you got a good brand) be UV resistant and not yellow over time. It will also look a lot nicer because the gloss doesn't dull the glitter.
The best sort of glue to use on pullips (and this is on pullips, not their accessories) is water based clear drying glue. In the US, Elmer's Glue is usually the most readily available. This is the sort of glue you use on wigs, eye chips and eye rubbers, eyelashes, etc. Because if you screw up just wash it off, even after it's dried, and you can start over good as new. Even if you saturate a wig with it and later take it off just put it through a warm wash with a little bit of soap or shampoo and you'll have no problems restoring it. The downside is that it takes a while to set. I use elastics to hold the wig down, set the eyelids under a lamp to dry faster, and eyechips and rubbers will usually stay in by themselves long enough for the glue to set.
Sometimes you need a fast acting ultra strong glue. These generally yellow over time and are not a good idea because if you screw up the only way to remove it is with a lot of scrubbing and acetone. Doing that will probably damage the doll, and won't guarantee the glue's removal. Fortunately, most of the things you will need a strong, fast bonding, permenant stick with will be repairs and those are mostly internal. Therefore, you don't need to worry about a little spill, or yellowing. Super glue is also good on accessories; fixing a shoe with a broken strap, for example.
Try not to use glue guns on pullips. Glue guns are great for bonding things that normally wouldn't bond well, like two round objects that don't have a lot of surface area between them. And it's good in making dollhouse furniture, when you are stretching fabric over bases you need something that won't damage the fabric but will dry quickly. But there's really nothing that it will do on pullips that something else won't do better with less mess and less damage. The thick glue residue, if you ever have to remove it, is extremely difficult to get off. You can hack and sand it off a pullip, but say you apply a wig. That wig is pretty doomed if you ever want to take it off and re-use it.
There's also the danger aspect of hot glue and the metal tip with pullip plastic. Pullip plastic isn't the best plastic. New dolls come out of the box with body melting from nothing more than one body part pressing against another body part. A glue gun tip will very likely melt the plastic for the same reason people pierce pullip ears with a heated needle. You don't need to touch the tip to the doll to apply the glue, but can you guarantee the tip won't touch the doll? It may be a light touch and not melt it, but it might not. Do you want to risk it? Great for accessories, not a good idea for dolls.
Did I forget anything? Let me know if I should add something to the list.