As far as the different signing systems in America, there are more than three, but all systems can basically fit into those three main categories. There are some that are "in-betweens."
SEE:
This includes SEE 1 and SEE 2. I always get them confused, but one means Signing Exact English and the other means Seeing Essential English. They are slightly different, but both use contextually inaccurate signs [i.e. my nose is RUN-ING (ASL sign for RUN with the affix ING after) This can be directed from the nose or signed in neutral space depending on which SEE is being used], use different handshapes for normal signs to show tense (HAS, HAVE, HAD, WERE, WAS), and use a lot of initialized signing (OCEAN-Same as ASL OCEAN, but signed with O-Hands). There is a two out of three rule for SEE 1 and 2. The three rules are:
- How the word sounds
- How the word is spelled
- What the word means
When deciding what sign to choose for SEE 1 or 2, this rule was invented (Yes "invented." Hearing people sat down and made up a language they thought would benefit Deaf children). If the word in question was BEAR and used in the sentence, "Please bear with me." one would look at the sound of the word bear, how it is spelled, and the meaning (notice that the meaning is the last in the list, making it the least important). Both the noun and verb form of bear sound and are spelled the same, but the meaning is different (as is the example with the word RUN), but that doesn't matter in SEE 1 or 2. Since two rules match, the noun BEAR will be signed instead of the verb BEAR. So the sentence will literally be "Please BEAR-ANIMAL with me" not "Please BE-PATIENT with me."
CASE:
Conceptually Accurate Signed English. Same as SEE 1 and 2 but it doesn't use the 2 out of 3 rule (thank goodness).
PSE:
Pigeon Signed English. This system uses ASL signs, almost completely conceptually accurate, no prefixes, and includes non-manual markers (expression, eyebrow movement, body shifts).
All of the above sign systems are considered MCE: Manually Coded English. This is because these languages were created with interference from outside the Deaf Community. Simultaneous Communication is also included in this category because it was not a natural way of expression when it was first used.
ASL:
The natural development of a language by people who use it everyday. It is very spacial, uses already natural movements for the anatomy of the human body (circular motions, not sharp jerky ones like some MCE), can express abstract ideas without using "words"... the list goes on.
I am by no means an expert, but one of my aspirations is to go to Gallaudet and get a degree in ASL Linguistics.
If anyone else has anything else to add, please do!