Gellan gum shows up in plant milks to keep added calcium and cocoa evenly suspended, and in vegan jellies and desserts as a gelling agent. Being a fermentation product, it is vegan, and it was quickly adopted as an agar substitute in both food and laboratory settings.
Xanthan gum (E415)Xanthan gum is a thickener grown by bacteria on sugar. It is almost always vegan, with a rare asterisk over what the bacteria were fed.
Agar (agar-agar, E406)Agar is the vegan answer to gelatin, a firm jelly set from seaweed. It has been doing this job in Asian kitchens for a very long time.
Carrageenan (E407)Carrageenan is a thickener pulled from red seaweed. It has its critics on other grounds, but the vegan question is not one of them.
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