There was no strict rule of wearing helmets in Red Army (and in memoirs Soviet soldiers were amused by US soldiers wearing helmets at all times). In early years of war lot's Soviet soldiers didn't want to wear helmets since they're were cumbersome, heavy and sometimes it was considered an 'unmanly cowardice' to wear a helmet.

It was changed in late years of war, because 'cowards with a helmet' tend to survive more.

Edit: And even in late years Red Army was no 100% helmet wearing.

The Soviets certainly had helmets. There were three common styles that were used during the war the Ssh-36, Ssh-39, and the Ssh-40 (which was functionally a simplified Ssh-39). Each model was named for the year in which it was first introduced. The 36 would have been rare by the time the war started. The 39 was the dominant type through about 1941 at which point the 40 became ubiquitous. You can see examples of them here.

What I don't know is production numbers or regulations on wearing the helmets. I'm hoping that someone else here can provide these additional details.