During WW2, the British had the Long range desert group, the SAS, Commandos, the SOE as well as airborne, the Americans had the airborne, the rangers and OSS, i would guess they had some more units but i cant remember more at the top of my head.
On the other hand the Germans had the Brandenburgers, fallschirmjäger, some SS units and if I am not mistaking Otto Skorzeny was in charge of some of them.
So did the Soviets have anything which one could compare in terms of special operations?
The Soviets had been pioneers in the development of airborne infantry warfare in the 1930s, but despite having the largest number of airborne combat formations at the outbreak of war — and the important potential role of airborne troops in their 'deep battle' doctrine — the Soviets mounted very few airborne operations during the course of the war.
The evolution of the spetsnaz ('special purpose'), Russian special forces, happened in parallel with the development of the airborne forces (VDV). The Soviets didn't fully adopt the commando/unconventional warfare model during the war: the first dedicated 'special forces' units were established in the late 1940s, under the auspices of the military intelligence service GRU. But the roots of spetsnaz are to be found in the partisan formations which were first created ad hoc to harry the German rear during Operation Barbarossa, and which later evolved into more organised behind-the-lines irregular warfare units.
The Stalinist purges of the late 1930s gutted both the Red Army and the state security organs. GRU defector Victor Suvorov claims that a by-product of the purges was the destruction of the USSR's ability to carry on unconventional warfare in the event of invasion:
During the pre-war years, in the areas of western military districts the intelligence directorates had extended the existing reserves of underground armies in case of the occupation of these areas by an enemy. Secret depots and stores of weapons and explosives had been established, radio sets had been secreted and refuges for partisans and intelligence officers had been set up. In the terror, all this was destroyed, and tens of thousands of trained partisans and saboteurs, ready to meet the enemy, were shot or perished in prisons and concentration camps. Military intelligence ceased to exist.
When the Germans invaded in 1941, panic set in. First, propagandists exhorted Soviet citizens in occupied areas to harass German forces wherever possible. Later, as the Red Army fought its desperate rearguard action, and gradually turned it into a large-scale counteroffensive, partisan operations were brought under central control with the creation of the Central Partisan Headquarters in early 1942. The NKVD, GRU and SMERSH (the dedicated counterintelligence service created in April 1943) quickly began deploying their own partisan units behind the lines. KGB statistics, quoted in Christopher Andrew and Vasili Mitrokhin's The Sword and the Shield, claim that the NKVD ran 2,222 'operational combat groups' behind German lines during the war.
The authors also express some skepticism, however, about the effectiveness of Soviet partisan operations — as have many others. Partisan groups throughout Europe were commonly riven by ideological and strategic disagreement, and in many areas spent a great deal of time engaged in internecine warfare. Perhaps more pressingly, there was as yet no coherent doctrine of unconventional warfare: the tactics and strategy of partisan formations behind the lines varied wildly.
In the Soviet case specifically, the chaotic state of Soviet intelligence and military command at the outbreak of war had a significant impact on the effectiveness of both their intelligence-gathering and sabotage/irregular warfare operations. The result was a mixed record: where Stavka could achieve effective command-and-control, and properly integrate partisan operations into wider manoeuvres — as in Operation Bagration in Belorussia in 1944 — they could have a striking impact on German logistics and communications infrastructure. But in many other cases, the partisans' really made little to no meaningful contribution to the war effort.
The Odessa partisans
(This is something of an aside from the core question, but it's a fascinating illustration of some of the point about the dubious wartime record of the Soviet secret services.)
Andrew and Mitrokhin highlight the dubious exploits of the NKVD in Odessa, immortalised in the 'Museum of Partisan Glory' in Odessa, as a bizarre example of the underwhelming wartime performance of the Soviet intelligence services — and of Soviet hagiography at work:
In October 1941, before the fall of Odessa, the NKVD in Moscow dispatched one Captain Molodtsov and six other officers to act as the kernel of a 'stay-behind' force in the city. Along with 13 officers from the Odessa branch of the NKVD, under a Lieutenant Kuznetsov, the NKVD set up a base of operations in the Odessa catacombs from which to launch raiding operations against the occupying forces. In fact, KGB records suggest that the Odessa and Moscow groups spent almost as much energy fighting among themselves as they did against the Germans.
In July 1942, Molodtsov was captured and executed. Kuznetsov then seized the Moscow detachment's weapons and placed them under arrest. Over the coming weeks, Kuznetsov executed 5 of the 6 Muscovites, and three of his own men. Finally, in October, the surviving Moscow chekist, Abramov, killed Kuznetsov. By this time, only three NKVD men were left alive — Abramov, Glushchenko and Litvinov.
At this point, it gets positively Shakespearean (if it wasn't already): Abramov and Glushchenko conspire to kill Litvinov; Glushchenko then reportedly murders Abramov (who apparently survived, and escaped to live in France), and eventually leaves the catacombs to go and hide out in his wife's Odessa apartment instead. Later, he returns to the tunnels for supplies and is killed when a grenade accidentally detonates.
Absent any (known) survivors, the adventures of the NKVD in Odessa — or rather, a myth that was created to replace the bizarre true story — were dressed up by Soviet propagandists as a fine example of spirited resistance, and a testament to the skill and dedication of the chekists. In actuality, it's a long way from being any kind of case study in effective special operations.
Sources
Edit: typo