What is the historical evidence behind the claim that the Soviet armies perpetrated mass rapes during their stay in Germany?

The initial question is itself shocking! Everyone in the former Soviet Zone of occupation knows about this and everyone knew someone who had been raped. Here is something to look at regarding "proof", in this case, eyewitnesses finally speaking out after 65 years!

http://www.washingtonpost.com/blogs/she-the-people/wp/2013/11/24/silenced-shame-hungarian-women-remember-wartime-rapes/

I want to provide some context for the developments in question. At the time in Germany abortions were prohibited unless as a result of rape, especially by "subhumans". If a woman claimed she was raped by a Russian, the abortion was conducted for free so to prevent racial pollution. This legislation remained in force for some time after the defeat of Germany. I say this just as a fact and not to allege that a part of the rape claims were done so to get an abortion permit, which is though quite possible.

Another circumstance is that there was a large number of ostarbeiters (forced workers), people freed from POW and concentration camps, pro-Germany East battalions composed of citizens of the USSR who collaborated with the Nazis, including Asians, Cossacks, Russian Vlasovites etc (which all had no discipline at the time), non-Russian Slavs such as Poles and other people whom the women could mistake for Russians. As such, any claim that a woman was raped by "a Russian" does not necessarily hint that she was raped by a soldier rather than, say a KZ escapee or anybody else.

There is another perspective on this, which is suggested by the historian Mark Solonin in a speculative 2009 article. The article is a long, wide-ranging and detailed one so I'll summarize the major points of his arguments, as I understand them. The wording is mine and so is the presentation:

1. There were indeed numerous atrocities committed against German civilians. This is well attested in German and Russian sources. For example Nemmersdorf. The details are a mess but the overall pattern is true.
2. The absolute majority of Soviet soldiers did not take part in thiese atrocities.
3. Solonin raises the quesiton of the demographic profile of the new recruits who joined the Soviet armies in Germany in late 1944-early 1945. Indeed, the four years of war (and the wasteful Soviet command style) took a heavy toll on the male population (an infantryman lasted on average 5 (five) months on the front line - after that he was either dead or invalided home). He correctly points out that many tens of thousands of soldiers were recruited form among the Soviet population that had endured the Nazi occupation. Generally speaking, this population was heavily brutalized by their experience of life under occupation and some of the younger men could well be expected to have lost whatever moral compass they had had while growing up under the occupation.
4. Here is where things get really tricky. Many of the people who were enlisted thus actually served the Germans before. Having undergone a casual background check (imagine how thorough could those checks be in the middle of a war, amidst total dislocation etc) they were drafted into the Red Army.
5. Here is a striking example: A Soviet colonel, V.V.Gil aka Rodionov aka Rodjanoff was captured early on in the war, joined the Germans and got his own command - the 1st SS Russian National Brigade. You can imagine what sort of service his troops saw. In the summer of 1943 Gil switched sides again and his unit was redesignated the 1st Antifascist Partisan Brigade (russian Wiki) as part of the Soviet forces!
6. Did such men (SS auxiliaries who switched sides) later become part of the Soviet occupying force in Germany? Nobody can tell for sure at the moment (closed archives, wartime chaos etc) but it is possible that some/many of the atrocities were committed by them. Add to this also the fact that it's enough to have, say, 5-10% rotten apples in a state of lax discipline (and Soviet discipline was very lax by then) to implicate the whole unit as such.
7. Now it gets really creepy: Solonin conjectures that the atrocities might have been implicitly encouraged by Stalin as part of his plan to de-Germanize East Prussia. He foresaw that his Allies would expect this territory to be pary of the future Germany but Stalin had his own designs on the territory. The atrocities were a way to get the scare the German civilian population and to force them to run away - as they did.
8. Here is one bit of evidence adduced to in support of the theory that the atrocities were encouraged from above - Stavka order No. 11072 dated April 20th, 1945 from Stalin demanded to improve the treatment of the German civilian population. Here's the crucial thing: it was specifically directed to the commanders of the 1st Belorussian and the 1st Ukrainian Fronts whose troops were in what was designated as the Soviet zone - whereas the 2nd and 3rd Belorussian Fronts, fighting in East Prussia and Pommerania were not given the order!

This is, as I said, highly conjectural and I can see some holes in the asrgument. But it does make sense.