The answer to this is complicated. On the Eastern Front, regular German soldiers commonly collaborated with SS forces (the Einsatzgruppen, the SD, the Orpo) in their operations. These operations included the mass murder of Jews and "partisans". Wehrmacht troops gave both logistical and direct support. In July 1941 Einsatzkommando 4b of Einsatzgruppe-C reported: "Armed forces surprisingly welcome hostility against the Jews". An Einsatzgruppen-A noted approvingly that by December 1941 "approximately 19,000 partisans and criminals, the majority Jews" had been shot by Wehrmacht troops in Belorussia alone. In occupied Greece too, members of the Abwehr (military intelligence) and Wehrmacht regularly collaborated with SS organisations on stopping resistance (one agreement was that the SD provided Wehrmacht troops with "communist" hostages to be executed in case of resistance attacks). German soldiers also agreed to help the SS with deportations of Greek Jews.
From my reading, it appears that much of the SS-Wehrmacht rivalry came from those higher in the hierarchy and not between regular soldiers.
Sources:
War of Extermination: The German Military in World War II
Inside Hitler's Greece: The Experience of Occupation 1941-44
I don't think that putting the "regular German soldier" as a different category from members of the SS is very useful. The Waffen-SS was a large armed force, and comprised regular soldiers as well who served in much the same way and in similar structures (albeit with different command chains, rank names, uniforms and so on) as the soldiers of Wehrmacht units did. The late Günther Grass is an example for this. Then you have the countless foreign conscripts that served in SS units, the SS recruited far more extensively from foreigners than the Wehrmacht did, so much that at the end of the war, about half of the Waffen-SS was made up of foreign troops. The SS was a vast organization that wasn't only concerned with the war of extermination. Neither was the Wehrmacht separate from this endeavour.
/u/blackjacksandhookers has already pointed out the complicity and the cooperation of regular Wehrmacht soldiers in the war of genocide and extinction that Wehrmacht and SS were waging. This can't be stressed enough, since a clean separation between these two branches is what lies at the heartof the 'clean Wehrmacht' myth, which sought to exculpate the 'regular' German soldier while shifting blame onto the 'evil' SS.
The regular, normal Wehrmacht soldier forced at gunpoint or threat of death to do atrocities by order of an SS-officer, or witnessing crimes perpetrated by the SS while remaining clean and innocent himself, is a common German/Austrian trope of postwar literature and film up until the late 1990s, and it still appears in German media from time to time. Basically, the Wehrmacht was just like any other army, and the SS did all the bad stuff.
/u/vonadler has also written about the anomisity of the Wehrmacht (mainly on the upper echelons) towards the SS here. Towards the end of the war, there was more animosity regarding the allocation of manpower, which was in ever dwindling supply, but more so towards the Gauleiter and their pet project 'Volkssturm'.
Just to give a bit of a different perspective on it.