I don't have much specific information about any given warship, but know several general points.
1. Ships with aviators on board (carriers, battleships, cruisers, and seaplane tenders only, I think) carried .38 revolvers to arm aircrew. While some aircrew were issued US 1911A1 .45 Automatics in shoulder holsters, it was the norm for aviators to be armed with S&W .38 Special "Victory Model" revolvers. As far as I know, this was an issue unique to aviators, at least on shipboard (some Shore Patrol got them too).
2). The firearms in the ship's locker controlled by the Master-at-Arms tended to be of the vintage of the ships first commission. So older ships often had REALLY old small arms--some of the ships commissioned in the First World War like 4-stack destroyers even had Spanish War issue small arms--Krag rifles, Colt 1894 .38s, and boarding cutlasses at the beginning of the war (because there was a shortage of small arms at the beginning of the First War, as well as the Second). I don't think any had Lee Navy 6mm rifles, but wouldn't be surprised if there were even a few of those left on the Phillipine and China station gunboats we captured from the Spanish in 1898 and were still using in 1941. And some of our Coast Guard cutters were REALLY old.
Later in the war these tended to get replaced by later weapons as these old ships were converted to other uses ("fast transports," DMSs, etc). But even then some WWI-issue weapons, like Colt 1917 .45 revolvers and "Enfield" 1917 rifles persisted all through the war.
The most modern small arms went to the Army infantry and airborne units first, then the Marine infantry, then everyone else, with Naval ships and the Coast Guard dead last. Not true in every case, I'm sure, but the general trend.