Loads. Two particular examples stand out in my mind.

• The Finns made extensive use of captured Soviet equipment from aircraft and tanks down to rifles and pistols.

• The German panzer arm took a pasting in Poland and the tank divisions that invaded France and the Low Countries had a lot of captured Czech tanks (designated Pz.38(t), where the t was for "tschechisch") and even some Polish tanks.

There are lots of examples of individual vehicles being captured and put into use, but it's often more for evaluation and testing. There are good reasons not to use enemy equipment in the front line: it's a recipe for friendly fire, your troops aren't trained on it, it might well not be compatible with your ammo.

I only know about tanks in this regard, and it's all excluding stuff like half-tracks and light armored vehicles like armored cars and the like.

Russia took to using German tanks for plugging holes in their lines. They liked the Panzer III and (especially) StuG III variants, as they were useful in infantry support, had an abundance of spare parts and ammo, and were pretty mechanically reliable. Panzer IVs were liked as well, Pz. 38s were useful, and the rest were taken without complaint. They weren't a big fan of the big cats like the Tigers and Panthers due to intrinsic mechanical problems, rarity of the vehicles, and lack of spare parts/ammo. There were some units that were able to get some long life out of the Panthers beyond their 150km maintenance range, but they eventually either dumped them for lagging behind or mechanical breakdown reasons.

What AFVs weren't pressed into frontline service were used for secondary fronts, training, or command vehicles. Some, like in the case of the Pz. III being turned into the SU-76i, were re-purposed into Soviet homebrewed vehicles. They supposedly used plenty of APCs, mobile artillery, etc., but I have no real concrete information on that. There's a table and some info on this page that seems pretty reliable:

http://wio.ru/tank/capt/capt.htm

Western allies didn't really use captured tanks. They usually had more M4s than they knew what to do with, the smaller Axis tanks weren't any better than the M4s, and big cat tanks were not common enough to really do anything with even if desired by command.

As a side note to your specific interest in Tigers, one big misconception is the Western Allies running into nothing but Panzer VI Tigers. They saw a few in North Africa, less in Italy, and America literally had three encounters with Tiger Is in all of Western Europe (Britain ran into quite a few in their sectors of Normandy due to more wide open spaces). They were rare and hard to deploy, so they had no spare parts or ammo, and had awful mechanical problems intrinsic to design.

Germany had a lot of gaps to plug after the invasions of Poland and France. Despite impressive victories, they lost considerable numbers of mechanized and motorized vehicles. They used a fairly large amount of French (the whole range of cavalry and infantry tanks) and Czechoslovakian tanks (mostly Panzer 38s, 35s were ignored for all i can tell), and used Soviet and American tanks in good numbers (I don't know about British tanks at all, but I assume they ran into enough Valentines in the Eastern Front).

Like the Russians, the Germans did what they could to rip up captured vehicles and throw German guns on them to turn them into mobile assault weapons or tank hunters. Tankettes and outdated lighter tanks were favorite targets for this treatment. They used lots of Soviet lights, mediums, heavies, and assault guns, but the use of rarer tanks suffered form the same lack of parts/ammo/knowhow that Allies ran into. They used M4s for battle and diversionary tactics (most famously in the Battle of the Bulge) when they could. These were nice mostly because they ran on the same gasoline that German tanks did, while most Soviet tanks ran on diesel.

I hope these answer your questions. A lot of this is thrown together from reading a hundred tank books over the years, and reading from a sparse supply of Russian archival documents, so if you'd like to know something specific, I may be able to point you in the right direction for more information.