I have been wondering all summer about the evolution of these, and I finally decided to investigate.

Problem:
Each spelling is considered to be correct, but not everyone agrees which among these three spellings is/are correct.

Explanation:
These three items mean the same thing, but the different spellings here represent historical variance.

For fun, I searched Google for each of the following (with the quotation marks, to avoid variations) and got about the indicated numbers of matches:

• “swimsuit” — spelled S-W-I-M-S-U-I-T — 27,700,000 matches
• “swim suit” — spelled S-W-I-M-SPACE-S-U-I-T — 2,110,000 matches
• “swimming suit” — spelled S-W-I-M-M-I-N-G-SPACE-S-U-I-T — 313,000 matches

This tells me that Web authors have used these three spellings by a compound ratio of 88.5-to-6.74-to-1.

This compound ratio seems to reflect the evolution of the compound noun “swimming suit” into the space-free noun “swimsuit”. From what I have seen, “bathing suit” was the original way to describe this piece of clothing. This seems to have morphed into “swimming suit”, especially in the USA. This seems to have morphed into “swim suit” without the I-N-G but still with a space before “suit”, which seems to have morphed into “swimsuit” with no space.

I believe that evolution of this word is consistent with my “Devolution toward Simpler” linguistic hypothesis. It is simpler to write or type “swim” than to write or type “swimming”, and it is simpler to omit the space than to include it.

Solution:
Use any these three forms, but realize that the most popular spelling today uses “swim” instead of “swimming” and joins “swim” with “suit” without a space.