Although the Larvotto would probably be my first choice for swimming in Monaco, it gets very crowded, and quite dirty, in the summer, and sometimes you may want an alternative.

You can pretty well swim along the whole coastline but here are the beaches and pools that are in Monaco or nearby (from East to West):

Le Beach
Private. France. Has a great pool, salt water, and a nice upmarket atmosphere. Great location. I think you've got to be a member or go with someone who is. Very cool. Is part of the 4 star Monte-Carlo Beach Hotel complex. Natural stone beach, hardly anyone uses it, very uncomfortable to walk on.

Small beach just past Le Sporting
Public. France. Is popular in the summer. Natural stone beach but uncomfortable to walk on. Not sure if it is has showers. Quieter and more natural than the Larvotto.

Le Meridien Beach Plaza
Monaco. Private. Right next to the Larvotto (East) - the beach is identical. Two open-air seawater pools, one for kids, one freshwater indoor pool. Good idea if you want to get away from the hordes or swim in a pool.

The Port Pool - La Piscine
Entry fee. Monaco. Popular in the summer. Large open-air pool. Salt-water. Have a look from outside. Skating in the winter.

skating

Outside of the new jetty in the port
Public. Monaco. It's not a beach but you can swim there. Nicely laid out on large concrete steps. You jump straight into very deep water, which is exhilarating but probably not for unconfident swimmers. I also saw some beautiful jellyfish that you don't get in the Larvotto. The significant drawback is that you're quite likely to get an oil slick floating past from one of the many boats and ships passing. This happened when I went, so we left.

Swimming outside port

Tiny beach under the Oceanographic Museum
Public. Monaco. I only noticed it this year when swimming outside the port jetty, but have not yet been. Access must be down steps from the Quai des Pêcheurs.

Stade Louis II pool
Fontvieille, Monaco. Entry fee. Freshwater. Olympic size. All sorts of rules: no bermuda shorts, no diving, must wear a cap, lane swimming only. But good for serious swimmers, especially in the winter.

Cap/Plage Marquet
Public. France. Just to the west of the Port of Cap d'Ail, which borders Fontvieille, the western end of Monaco. It's a decent beach: natural, so stones, but pretty popular and much less built up than Monaco. I'm pretty sure it has showers, there are restaurants bordering it, and a tennis club.

Also (unchecked)
Public indoor freshwater pool - 7 av St Charles
Hotel Mirabeau - open air seawater
Les Thermes - indoor seawater