I'm thinking Amn would be the obvious answer on where you might find bugbears. But I'm having trouble thinking of any others. Cormyr definitely seems like its cities would be free of them.
Orcs are excepted in the Lands of Intrigue, and inter-marry with low-born humans frequently. In Erlkazar, a tribe of Ogres are even in the employ of the gov't to watch over the mountains. Given these facts, I can see well-behaved Goblinoids being excepted there as well.
Several tribes of Goblins and Orcs get along with their human neighbors in the Vilhon Reach, and according to the Everis Cale series, Gnolls as well. Once again, I can't see why Bugbears wouldn't also be welcome, as long as they minded their manners.
Thanks to Zhentil Keep, Orcs and other 'monstrous types' are seen frequently in the Moonsea cities (for the most part), so once again, no problem with Bugbears there.
The Unapproachable East is rather approachable, from a Goblin point-of-view. Aside from the fact that Thay makes heavy usage of Humanoid armies and slaves, we also have Orcish 'heroes', thanks to the Tuigan wars, living in the region. It is entirely possible for civilized Bugbears to make a place for themselves in that region (except, perhaps, in Aglarond).
Not really seeing it in the North, except maybe for a few exceptional ones in Waterdeep (ANYTHING can be found there), and the same goes for the south (except for 1e/2e/3e Dambrath, since Drow are famous for using Goblinoids as well).
No chance in seeing that in the Eastern Heartlands - Cormyr, The Dalelands, and Comanthor Elves all kill Bugbears on sight. In 4e, I suppose anything is possible though, and their may be some in Sembia thanks to the Netherease (although there is nothing in canon to say they used Bugbears at any point before, but they did love to use slaves, including Goblins).
Outside of Faerûn proper - in Zakhara, Humans, Demi-humans, and even Humanoids ALL live together peacefully in a unified culture. Weird, I know. In Kara-Tur, there are no known canon Bugbears, and the only Goblins are quite a bit different from their western counterparts. There are Gnolls, however.
Non-Canon:
For 4e, I have placed a large kingdom of Oriental Bugbears in the Plain of Horses, on the Kara-Tur/Hordelands border. They have history going back to 3e (and the past), and I am tentatively tying them to the Kalmyk invasions of the Hordelands (which is canon, but hard to place in the timeline). As of right now, I'm saying the original Kalmyk civilization was human, and spread all around the Great Ice Sea. Unfortunately, they decided to breed the Bugbears in the mountains as a 'Thrall Race', serving at first as servants, and then later as soldiers. The usual happened, and decadence lead to their slaves over-throwing them and driving them south (the Gur), while the Bugbears (now civilized, thanks to the Kalmyk), maintained the old human Kingdom for centuries. That Kindom fell after a massive period of expansion, wherein they took large parts of the Taan (Hordelands) and even northern Shou-Lung (these were the canon Kalmyk invasions). The cause of the collapse was typical for such quickly-expanding cultures - eventually the leader died and the various provinces turned on one-another, thus ending the glorious (and brief) Kalmyk Empire.
Thats just a brief synopsis - I tie their rise to the fall of the Raumathari, the Iron Demon of Tros, and the Lost Land of Tsharoon.
On the Dragon Reach, Westgate is the most likely city to allow "civilized goblinoids" (i.e. those employed as mercenaries, etc) to live within the walls of the city. With the expansion of Cormyr and the increased danger on the Sea of Fallen Stars (from Xxiphu (sp?) and renegade pirates/privateers in the employ of Sembia (Netheril) I can see them taking all the ready swords they can hire.
The city of Airspur in Chessenta (classic FR) is inhabited by humans, orcs and half-orcs. The orcs were refugees of tribal war in the mountains, they came to Airspur, and kicked out the evil priests of Bhaelros (Talos), moved in and everyone lived happily ever after (after some initial human-orc disputes, and until the genasi arrived, grr). The orcs, both mountain and city, trade and deal peacefully with the humans, when they're not warring.
At least one ogre can be found working in Procampur in the Vast, as the apprentice to a dwarven printer, no less. (This was in the short story "Patronage", with absolutely nothing special about this ogre.) I expanded it IMG to include ogre guards employed by the city militia, ala the trolls in Ankh-Morpork. (Elsewhere in the Vast, orcs and goblinoids are pretty much shoot-on-sight however).
The Utter East, in the Blood & Magic computer game, has goblinoids in various capacities. The so-called "gnomes" are in fact green and look like D&D goblins, they seem to work as artificers and tinkers. The so-called "goblins" are also green, but are more like D&D orcs, hobgoblins or bugbears. They were once part of the evil Kingdom Of Nix, but the human nations treated them with honour and diplomacy following the death of their fiendish (?) boss.
-The Border Kingdoms already have a few cities that are, more or less, "monster cities". Oeble is the most "famous/infamous" of them, I am thinking.
Is there any supplement where detailed information on the Border Kingdoms can be found?
That said, Power Of Faerun began to detail them, and Ed Greenwood did a series of articles about them on the WotC website, though he only got partway through the alphabet before 4th edition hit.
Is there any supplement where detailed information on the Border Kingdoms can be found?
-Power of Faerûn has a partial index of some of the kingdoms there. But, the place was designed in the first place to be filled with minor kings and queens with realms that build up and fold on a regular basis. What's there today isn't necessarily going to be there tomorrow.