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PQRST is definitely a leader in publishing new and original Battleship variants.
Here are two such variants from their 13th PQRST Puzzle Championship.
These variants include:
Square Box,
and PQRST Letters.

Here’s a full analysis of Fathom It!’s hard-level #190 puzzle.
The analysis shows the battleship must reside along one specific row.
Once that is established, the correct positioning is quick to follow. The rest is, as they say, just technique.

The LEGUP project at the Rensselaer Department of Cognitive Science is an undergraduate research project
whose objective is “to research and categorize various grid-based logic puzzles, with the intention of
creating a formal system of ‘grid logic’ applicable to various puzzles of different formats”.

One of the logic puzzles they are tackling is Solitaire Battleships. Check
out the link on the MVS
web site for more information.

Diogen, the Russian puzzle site, always includes a Battleship variant in its puzzle championships.
We’ve added six new battleship variants from their 2003, 2004, and 2005 puzzle tournaments.
These variants include:
Triple Fleet,
Observers,
Boarding,
Shuffled,
Halves,
and Laser
Battleship variants.

The Russian website, http://kig.tvpark.ua/, has PDF versions of a weekly puzzle magazine. Half of the puzzles are logic puzzles that do not require knowledge of Russian. They have over a year of back-issues available for free download. Each issue has an Battleship puzzle that is standard except in size — it’s 11×11 with fifteen ships — five 1’s, four 2’s, three 3’s two 4’s and one 5.

Many thanks to Edward Jackman for pointing this out.

We’ve added a new battleship variant from Diogen’s second-round 2005 puzzle tournament entitled “Aiming Battleships”.