There was a great deal of non-combat cooperation: American and British personnel worked closely with Soviet counterparts to set up the Lend-Lease program, which was a monumental logistical effort. Soviet personnel trained in Alaska on American ships and planes before bringing them over the Soviet border.

I am not aware of any planned joint ground operations, although this may have happened in an ad hoc way; I do know that downed British and American pilots fought behind Axis lines with Soviet-backed partisans (primarily in Yugoslavia and Romania), so it's very possible that a handful of Americans and Soviets fought side by side before 1945.

Further, the U.S. military tried to establish a military mission to the Soviets to provide advice and expertise: this was refused due to security concerns and the would-be advisers went home after stewing in Tehran for months.

Later in the war, the USAAF launched Operation Frantic. At the December 1943 Tehran Conference, Roosevelt agreed to supply U.S. bombers to support the summer 1944 Soviet offensive. In February 1944, U.S. support personnel began upgrading the Soviet airfield at Poltava. On June 2, 1944, a few flights of B-17s took off from Italy, hit targets in Hungary, and then landed at Poltava. Over the next couple of weeks, the B-17s struck targets in Romania and Germany. A German recon plane shadowed the bombers back to Poltava - it had been a Luftwaffe base the previous year, so the Germans knew well how and where to knock the base out of commission. On June 22, German bombers destroyed 47 of the 73 B-17s on the ground and damaged most of the rest.

With only nine operational bombers, Operation Frantic was drastically downscaled but attacks continued through September. At that point, the progress of the war in France and the difficulties of working with the Soviets led the USAAF to suspend operations. By the time winter weather lifted, the frontline was too far west for Poltava to be effective, and no Operation Frantic flights were launched in 1945.

I don't know as much about this, but small Free French and British air units fought in combat on the Eastern Front.

The British RAF No 151 Wing was sent to Murmansk with 40 Hurricanes and 550 personnel in September of 1941. The force was to provide immediate air cover to protect Murmansk and then to train Soviet Pilots to use the Hurricane. The fighters only shot down 15 German Aircraft but more importantly the air cover they provided prevented the German dive bombers from harassing Soviet ground forces. The wing headed home in Late November 1941.