So I looked up つける and saw it can be written as:
And they all seem to have the exact same definition:
to attach, to join, to add, to append, etc. So are these "spellings" interchangeable or is there some difference?
As is usual with homophonous kanjis, there is a general one, which, in this case, is 付ける. That means, in general, the other kanjis can be replaced with this one, but not the other way around. 着ける is used when the attachment is along a surface, especially in wearing clothes (that is, the clothes touch the skin along surfaces). 附ける means to append, and I think it is an archaic kanji. Most often, you can find it in combnations like 附属 or 附設 (archaic form of 付属 and 付設), especially in proper names like 久留米大学附設 高等学校・中学校.