Where did Sauron go after his defeat in the Second Age?

Question

Sauron was defeated in the War of the Last Alliance and his spirit fled from his body. Where did he flee away and hide for thousands of years? Are there any details as to where he went or is it just plainly 'his spirit left his body and didn't return until the Third Age'?

Answer 1

No, this is never discussed in any detail

Gandalf tells us early in Fellowship only that Sauron's spirit "fled" (emphasis mine):

Isildur Elendil’s son cut the Ring from Sauron’s hand and took it for his own. Then Sauron was vanquished and his spirit fled and was hidden for long years, until his shadow took shape again in Mirkwood.

^{Fellowship of the Ring Book I Chapter 2: "The Shadow of the Past"}

But as far as where he went specifically, all we're told is one line in The Silmarillion (emphasis mine):

Then Sauron was for that time vanquished, and he forsook his body, and his spirit fled far away and hid in waste places; and he took no visible shape again for many long years.

^{The Silmarillion V Of the Rings of Power and the Third Age}

Answer 2

As a side-note to the previous answer, it can be noted that Sauron lost his ability to transform into a "charismatic being" when Eru drowned Númenor under the sea. He was only able to take shape again because he wore his Ring, in which he imbued much of his power and strength. When separated from his Ring in the "War of the last Alliance", it is plausible he lost yet another aspect of himself.

Without the ring, it certainly took Sauron many hundreds of years to "evolve" from "tingling annoying mist" to "evil spirit with some sort of physical appearance" then to "necromancer of Dol Guldur".

Except for the very last hundred years, his dark brooding was certainly close to boredom and nothing much of interest. As for possible in-verse sigthings, knowing that the Council discovered the truth behind Dol Guldur very late and that Orcs / the evil side are not fond of keeping history records, it is unlikely to find much evidence on Sauron's whereabout's during his "recovery" :)