First lets take on ほとんど～ない and めったに～ない
The big difference between the two can be expressed with these two keywords: 程度 and 頻度.
ほとんど～ない expresses 程度 (you could think of it as "degree" or "extent") and 頻度 (frequency) (credit @sawa for pointing out ほとんど has both)
めったに～ない expresses 頻度 only (or 回数の多さ, you could think of it as "frequency" or "how often something happens (or is performed)).
Also, めった in 漢字 is 滅多. Notice the relationship with 多?
Lets look at some examples:
漢字がほとんど読めない
彼の声がほとんど聞こえない
In the above examples, the ほとんど cannot be replaced with めったに. Do you see why? The concept of "how often" for the ability to read 漢字 and ability to hear someones voice doesn't make much sense.
However, lets look at an example were both can be used.
テレビをほとんど見ない
テレビをめったに見ない
Both can be used here. However, I feel that ほとんど～ない is more common (Googling both phrases gave me a lot more hits for テレビをほとんど見ない.) There also may be other subtle differences, however I could not find conclusive evidence at this time (if anyone has any insight please let me know!).
Also,
めったに見ない車
When I read the above (the action of seeing), I think めったに～ない is better here (unfortunately that might be a personal opinion).
Now, lets add まれ to the bunch. まれ falls under the same category as めったに, in that it expresses frequency. The difference is in the usage (the meaning is exactly the same as めったに見ない).
まれに見ない // Not Japanese
まれに見る // Okay
まれ technically is already a negative expression meaning 実現・存在することが非常に少ないさま。また、数少なくて珍しいさま (reference), so you can't use it with a negative as in the first example because it would become a double negative.
