What happened to most of the members of the SS after the war? Were they punished?
The Schutzstaffel was a paramilitary unit created by the Nazi party. After the Nazi party came to power and the Nazi party effectively became the state, the SS as a result also became much less of a paramilitary unit, and instead become just a second military unit, parallel to the Wehrmacht.
The difference between the Wehrmacht and the SS was that SS was seen as an elite force, and was also populated by pure "aryans" who at least claimed to believe in the national socialist ideology.
As such, most of the members were not punished, because you were not punished after the war for being "aryan" or being a Nazi, for pretty obvious reasons. So yes, most of the over one million members of the SS were not punished after the war. The exception was those that were convicted of war crimes.
The Schutzstaffel was a major paramilitary organization under Adolf Hitler and the Nazi Party. When Heinrich Himmler became leader of SS(1925.) it grew up to largest and most powerful organisation in Third Reich. There were lot of SS divisions in WW2, and SS members commited lot of crimes. On Nuremberg trials SS was declared criminal organization by International Military Tribunal, and banned in Germany after 1945, but most members of that organization were not punished.
There were a number of major possibilities and minor possibilities. They were:
(1) If you were captured by the Soviets you were summarily executed. They had big rewards out, so being anywhere in the Soviet zone, including Poland, Romania or the Balkans was virtually certain death.
(2) If you were captured by Tito, guaranteed death.
(3) If you were captured in the American zone, then it depended on whether you were in an eastern or western unit. If you were in a western unit, you were thrown in jail (see below). If you were in an eastern unit, you were turned over to the Soviets (see 1 above).
(4) If you were captured in the British zone, you were thrown in jail (see below).
(5) If you were captured in the French zone, in the Saar Protectorate or in France proper, then a special unit of the Free French Army tried you secretely. If they found you had any crimes against the French you were shot and buried secretely. If not, then there were two possibilities: you could choose to remain in France, or to return to Germany. If you choose return, you were turned over to the British and thrown into jail (see below).
(6) You could escape to Spain. You could then try to hide in Spain or move on to South America via Portugal. Remaining in Spain or Portugal was risky because the French had special units that hunted and killed anybody on their bad list, and their bad list was quite long.
(7) If you got jailed, then what happened is that the British or Americans "interrogated" you which means they beat you with rubber hoses. The more they thought you were hiding, the more you got beaten. Other tricks were used as well, like stool pigeons. If evidence was found against you, or you confessed to a war crime, you got bumped up to war criminal status and were tried at Nuremberg. About 5000-10,000 got convicted overall, depending how you count. If no evidence was found against you, you were released in anywhere from 6 months to 4 years, depending on re-education. If you renounced Nazi-ism and acted like a good British/American, you got out earlier, otherwise later. For SS, release in under a year was very rare. Like the French, the British gave them the choice of becoming British. For SS from eastern zones, they always chose to become British, because otherwise they would be released to the Poles or the Soviets (see 1 above).
Not a complete answer but some (particularly those who were in a position to be charged with war crimes) fled, a lot of them reportedly to South American countries. I've heard more than one person tell me that some South American mountainous regions reminds one of Europe's (inc. Germany) mountains. Although the primary reason would be that South America was remote and underdeveloped and they probably thought that no one would look for them over there.
Argentina had a couple of well-known ones, including Eichmann who was eventually captured by Israeli Mossad agents and taken to face trial in Jerusalem, where he was hanged.
Some members of the SS also joined the French Foreign Legion and fought in the French Indochina war. There are rumors that one regiment of the French Foreign Legion sang in German in the years after the war, reportedly because that regiment had a high influx of ex-German soldiers, including SS (although I believe the singing bit is likely urban legend, doubt the French military would have accepted it).
Ref for the French Foreign Legion bit: