I recently studied a course on the history of the Soviet Union as an elective in college and remembered reading somewhere that the illegality of drugs was one of the few areas where both the US and the Russians were able to agree during the Cold War. Despite this however, knowing what we do now about the widespread corruption and general inefficiency of the government in the Soviet Union, was there a black market for drugs in the nation and what drugs would have been the most common?
Short answer: weren't prevalent. Hemp, poppies were grown in the wild and cultivated widely - and not for drugs.
Take a look at this thread: http://www.reddit.com/r/AskHistorians/comments/1rw4il/how_common_was_marijuana_use_in_the_ussr_if_you/
However, drug abuse existed - as well as harsh criminal punishment for selling illegal substances.
widespread corruption and general inefficiency of the government in the Soviet Union
corruption really started to pick up in late 70s. As for "general inefficiency" - that's gross oversimplification. Some branches were quite efficient - soviet space program, for example. "Soyuz" is cheap, simple and reliable, used to this day - for example. Electrification plan GOELRO, public education program - examples of successful state programs. Agriculture and production of consumer goods are examples of two areas with huge issues.
was there a black market for drugs in the nation and what drugs would have been the most common?
Most certainly - it was much smaller comparing to present. Marijuana and amphetamines. Opiates.
Amphetamines and opiates were derived from medical supplies - see [1]. But most of drug users were using barter. Money didn't mean that much in USSR.
Sources
Russian - Interview with Bayan Shiryanov Bayan Shiryanov (real name is Kirill Vorobyev) is an author of several autobiographical novels about narcotic 'counter-culture' in late USSR/Russia of 90s.
JSTOR article - Mary Schaeffer Conroy. Abuse of Drugs other than Alcohol and Tobacco in the Soviet Union - Soviet Studies Vol. 42, No. 3 (Jul., 1990), pp. 447-480 Article shows, that drug abuse rate was low and concentrated in several hotspots. War in Afghanistan has led to spike in illegal drugs consumption.