● 1 Pre-Crisis Supergirl. Cousin of Superman. First appeared in Action Comics #252. Died during Crisis.
● 2 Matrix/Linda Danvers. Originally an artificial life form from an alternate universe that thought it was Lana Lang, merged with Linda Danvers when she was dying to form a composite being which later manifested an aspect called the Earth Angel. After all sorts of complicated adventures the Earth Angel aspect left Linda but she retained many of her powers. Currently retired.
● 3 Pre-Crisis Supergirl again. Appeared in the final story arc of Supergirl 2 comic (v4). Sent back to become Supergirl 1.
● 4 Cir-El. Claimed to be Superman's daughter from the future but was in fact a semi-clone created by some strange bad guys to resurrect Brainiac (no, it doesn't make any sense even if you read the comics). Presumed dead on no evidence at all.
● 5. Power Girl. First appeared Pre-Crisis as the Earth 2 Supergirl. Post Crisis was given several different origins which never really took. Real origin currently being addressed in JSA Classified. She may or may not now have some Kryptonian connection.
● 6. Animated Supergirl. Only appears in Superman and Justice League TV shows and comics based on those shows, she wears a costume that was copied by Supergirl 2 around v4 issue #50.
● 7. New improved kryptonian Supergirl. The current model. Basically a rerun of Supergirl 1 updated to present day.
Not forgetting all the alternative Supergirls that cameoed in the last arc of Supergirl 2's comic (v4). The story was about a villain who had been imprisoned by Supergirl in the future and so he was hunting Supergirls through all the alternative dimensions and killing them. He eventually recognised Supergirl #2 as the one who would defeat him.
And just to be completely thorough:
● 1a. Pre-Crisis Bizarro Supergirl. Appeared only in one story in Superman #140. Died after cuddling up to some blue kryptonite.
● 1b-g. Six mini-Supergirl clones that later bonded to form one full size Supergirl clone in v2. I don't honestly remember what happened to her, er, them.
● 2a. Bizarro Supergirl 2. First appeared in Supergirl v4 #62. Still alive as far as I know but has not been seen in some time.
And in case you were wondering how the current run can be volume 5, it goes like this:
v1. Supergirl 1, early 1970's
v2. Supergirl 1 again, late 70's
v3. Supergirl 2, 4 issue mini-series, mid-90's
v4. Supergirl 2, 1996 - 2003.