It is likely that this is due to my poor understanding, but why are there so many names for「すし」? I think for the most part I have only seen the first three (寿司、鮨、鮓). For whatever reason in some restaurants in America there are these other compositions listed.
Something tells me that their is some nuance between them or not all of them are in use. What are the differences between these words, and why are there so many?
First, we can split your examples in to two categories: Words exisiting before the Edo period and 当て字 existing after the Edo period:
当て字 existing after the Edo period：
寿司、寿斗※、寿し and 壽司※
※:I have lost confidence in where these 当て字 come from. I found a couple sites that claim that 寿斗, etc. originate from the Edo period. However, the sites do not have anything to back this claim up, so I cannot find any conclusive evidence.
Words exisiting before the Edo period:
鮨 and 鮓
鮓 (originally comes from 酸し (see Dono's comment) meaning "sour") refers to fish that has been salted or put into [糟]{かす} and fermented. Originally, this was done to preserve the fish (before refrigerators were evented) and is arguably considered to be the most "correct" version. [鮒鮓]{ふなずし}、[鮎鮓]{あゆずし}、and [鯖鮓]{さばずし} would all fit into this category.
鮨 originally referred to [魚]{さかな}の[塩辛]{しおから} from China, however it soon became mixed with 鮓 and both have been used interchangeably for many centuries.
寿司 was an 当て字 developed in the Edo period to mean [寿]{ことぶき}を[司]{つかさど}る or [寿司]{じゅし or よごと} as it was a special food used in celebrations, etc.
寿し would be the same as above with 司 replaced with し.
寿斗 is an 当て字 also. However, I could not find conclusive evidence on the use of 斗, I suspect it has to do with Masu and its use for drinking sake.
壽司 is just a variation of 寿司, as 壽 is just a variation of 寿.
Nowadays in Japan, I only see 寿司 and すし most often, and very rarely 鮨 and 鮓 (probably used more on expensive restaurants). I have never seen 寿斗, etc. in Japan (as also mentioned by sawa that they aren't used in modern Japanese), but this is my own personal experience. Also, the reason there are so many is because its purpose has changed overtime. Originally it referred to fish that was preserved so it wouldn't go bad, next it became known for a special food used in celebrations, etc., and nowadays sushi is so common and widespread, that it slowly is becoming more of an everyday food (although it still is considered somewhat special and is eaten during the holidays, etc.).
If you collect all the written forms of sushi you can find in America, you will find writings that are not Japanese because the majority of sushi restaurants in America are operated by Chinese or Korean people.
Use 寿司 the most commonly. Sometimes use 鮨 on Sushi bentobox. Other complicated kanji/ateji might be used because it would look cool, out of ordinary writing.
