As World War Two progressed, Hitler ordered the formation of more and more "SS" or "Schutzstaffel" divisions.
I am aware that these units often had priority for reinforcements and supplies, but how effective were they in combat compared to non-SS formations? Did they have better/more training for example?
Was it cost-effective for Germany to have so many SS units later in the war?
I am aware that these units often had priority for reinforcements and supplies, but how effective were they in combat compared to non-SS formations?
It depends on which formations are being compared. The best Waffen-SS units (e.g. Das Reich, LSSAH, Wiking) were every bit as good as the best Heer units (Grossdeutschland, Lehr, Deutsches Afrika Korps). The worst ran away as well as the worst Heer units (e.g. SS Gebirgsdivision Nord at Salla in 1941). On the average the Waffen-SS were probably more effective, but I only say that because taken in aggregate the Waffen-SS was on average more heavily motorized than the largely horse-drawn Heer. Earlier in the war the fact that Waffen-SS members were volunteers rather than conscripts would have had a significant impact.
Did they have better/more training for example?
They had different training. I don't have my reference book at hand, but my recollection is that the training centre in Bad Tolz focused on combat skills rather than Prussian-style robotic discipline. For example, the Waffen-SS pioneered live-fire exercises as a standard training method.
Was it cost-effective for Germany to have so many SS units later in the war?
When you look at the overlaps in the bureaucracy and military forces of Nazi Germany it becomes clear that cost-effectiveness was not a real concern, political factors took pride of place. The expansion of the Waffen-SS to the size it achieved is better explained in terms of (a) Hitler wanting a Praetorian Guard to protect the regime from a military coup d'etat, and (b) Himmler wanting to be the power behind the throne.
Heer had an excellent tradition of troop replenishment (training reinforcements before throwing them into battle). However, as the war progressed, Heer started to founder under the mounting pressure of numeric and material superiority of the Red Army (also, the losses among battle veterans mounted and the troops became weaker).
Hitler, faced with a cognitive dissonance (his Ubermenschen are being defeated by Untermenschen), chose to blame the Heer (from privates to Generals) and decided to form Waffen SS (which were similar to Guard troops in other countries).
Heer resented this because it starved it of men and materiel and reduced its prestige. Thus the German Generals tend to claim in their memoirs that Waffen SS were poorly trained (probably true), better equipped (probably true), less battle-effective (hard to verify), and more prone to war crimes (not necessarily true - not because Waffen SS was innocent, but because Heer was just as guilty).
Incidentally, this was not the only case of German land forces fragmentation, the other being Luftwaffe Field Divisions.
No matter what Germany would have done after 1942 could have changed the final outcome. They could have been more efficient, but the end result would have been the same.
Original SS (like Leibstandarte) units were better than common Herr divisions, because they had not only better weapons but also because they had more (volunteer) men. These units were quite effective in combat because they were more risk-takers, but the cost was that they had more losses as well.
I'd say that compared to elite units of german Army, like 1st Fallschirmjäger or Großdeutschland, elite SS units were similar in prestige and capabilities.
As Doug B says, later increase of SS units at the end of war was due to political reasons, not to improve effectiveness. Besides, all volunteers of foreign countries were sent to SS units (except for spanish volunteers).
