The Soviet navy never got to become as big as the British or even German.
Up to the 2nd World war Soviets largest navy ships were the ones made before 1917. The most famous example is Soviet navy flagship October Revolution, which was in fact modernized imperial battleship Gangut, built in 1911. But most imperial Russian battleships in the Soviet times were allowed to come to dilapidated condition and mostly sold to scrap.
Th Soviets started their navy history with the Brest treaty they signed with the Germans. The treaty was almost unconditional surrender. It is enough to say that Soviets even agreed to return to Turkey the territories ceded to Russia as a result of Russo-Turkish wars in the 1870s! Part of the conditions of the treaty was to hand over the whole Russian Black sea fleet to Germany, while WWI was still ongoing. This was so ridiculous, that the crews of the most Black sea navy ships refused to do so and instead blew up their ships. Later the Soviet propaganda spinned it the way that Soviets agreed to hand over the ships just to deceive the Germans, and instead ordered the crews to blow them up - totally ridiculous nonsense.
Even before that, in March 1917, the first thing what happened in the Russian navy, was a horrible massacre of navy officers by the sailors, in which 76 were killed, including the Commander of the Baltic fleet, admiral Nepenin (who ironically was liberal-minded and supported the Revolution). This tradition continued in later years. In 1917-1920, Soviets killed 28 out of 106 imperial Russian admirals, most of them without real reasons, just for being officers in the former navy.
Not a very good way to start building a navy, is it?
It is significant as well, that the Soviets could never even produce the cannons similar in size to the ones produced by imperial Russia in large numbers. Most large-caliber guns from the old battleships were scavenged and used throughout WWII as mobile artillery units, placed on railroad platforms, or even mounted on the strategic heights together with the old ship turret. One example is one of the Soviet batteries made with an imperial battleship turret mounted on one of the Russia's Pacific islands (Russkiy island), this battery was in service up until 1997, i.e. for 90 years since the cannons were manufactured! This is one of them, so-called Voroshilov battery. The cannons used are taken from the 1909 battleship Poltava in 1939. In this case, they reused the whole turret together with system of ammo feeding, targeting, etc.:
I think they must have been the oldest in-service cannons in the world, when the battery was finally abandoned.
The original battleship Poltava:
You can still see several cannons from imperial battleships in Moscow in the Victory park:
As far as I know, the Soviets have not built a single battleship before WWII, their largest ships were all modernized imperial ships, i.e. at least 20 years old. No wonder that if Soviets could not even produce cannons for battleships, they could not really build any battleships.
This to me explains why the Soviet navy was so insignificant during WWII.
But as others said, Soviet navy did see some action on the Black sea. There is even a decent computer game dedicated to this - Men of War: The Red Tide.