1) Undead: In this case, civilization is the cause of the monsters and not the cure. The evil that men do to one another, as well as the simple fact that dead bodies are piling up in places, leads to the dead walking around. But, so long as you don't go into the cemetary/ancient tomb/haunted house at night, you are never going to notice. Meanwhile, the various clergy are there to keep them at bay.
2) Fey: In this case, the monster is invisibly inhabiting the same space as the civilization. There are sprites in the copse of trees adjacent to the flower garden. There are brownies in the attic of the old home. Maybe there is even a redcap living in the marshes below the old mill. Most of the monsters are rather harmless, and so long as you don't go off alone, you'll probably never encounter your neighbors.
3) Spirits: Again, in this case, the monster is invisibly inhabiting the same space as the civilization. There various minor spirits haunting the area. Most of the time these stay off in the spirit world and there interferance cannot be directly observed.
4) Animals: These are inhabiting the edges of civilization and occassionally invading it. Most of the time they aren't terribly aggressive, and mostly you are only going to encounter these in the context of protecting livestock.
5) Beasts: Griffins, Hippogrifs, Owlbears, Pegasi, ect. are just rarer and more dangerous animals. Usually these live 20 or more miles away from civilization, and only occassionally stray into civilized lands precisely because its usually a pretty dangerous thing to do that. Both parties usually end up suffering for it.
6) Outsiders: This is another case in which the civilization is the reason for the monsters existance. Misguided summoners represent a danger for the society. Outsiders will also be naturally attracted to places (and events) of extreme good or evil. Most of the time, these act rather similarly to any other 'spirit' creature, and spend thier time ethereal or otherwise subtly influencing events.
7) Oozes: Any isolated moist dark area with a moderate ammount of material being washed into it can become an ooze home. Since oozes are typically very quiet creatures and perfectly happy not moving unless they have to, they can live quite happily in an inhabited region without being noticed for a rather long time.
8) Plants: Like oozes except that the area doesn't need to be dark.
9) Dragons: Like beasts only bigger and more dangerous, and thus require a larger territory to call thier home. Still, just because you've never seen a dragon doesn't mean that the nearest one isn't a scant five days march through the wilderness away from you - or a half days flight for it. Smallish dragons are basically just beasts, and typically don't survive straying into civilization for long. Medium sized dragons are usually smart enough to stay out of civilization's way lest they attract heroes and have thier beautifully long lives cut short. Big dragons are rare, occupy the most disable territory (at least as far as dragons are concerned) far from large populations of humanoids, and spend most of thier time torpid and dreaming dark dreams about gold and blood. When they do stray into civilization to rob it, they generally win the encounter, but they are usually smart enough to not show up in the same place every year lest the humanoids go from terrified to vengeful.
10) Humanoids: Your neighbors. You know about them. They know about you. You try to leave each other alone, and they mostly leave you alone - although a few intrepid (or stupid) souls may try to rob or sell something to the other one (or both). In some cases, there is something like an alliance or even in rare cases a friendship.
11) Giants: Like humanoids, only smaller quantities in the same space. Stupid giants tend not to be terrified of humanoids and so end up dead before learning that you can't always judge things by thier size. Likewise, stupid humanoids tend not to be terrified of giants, and so end up as food.
12) Shapechangers: Invisibly inhabiting civilization, but typically cause a much bigger stir whenever they are discovered than fey, spirits, or even outsiders. This is because people are really really bothered when things can look like things that they are not, and in particular when things can impersonate 'people'. Moreover, shapechangers tend not to be as good of neighbors as fey or even spirits, and tend to take advantage of their abiity to blend into a population to commit every sort of murder and atrocity. The result is that shapechangers and civilization leave in a state of perpetual war. The presence of detected shapechangers causes a responce that actually far exceeds the actual number of shapechangers that actually exist, so that from the prespective of a paranoid society a very few shapechangers represent a major and common threat even if more people are actually threatened by beasts, animals, humanoids, etc. On the other hand, considering the social consequences of failing to stop the spread of lycanthropy, this level of responce might not be irrational afterall.
13) People: Lots of opponents are simply evil persons (whether human, elves, dwarves, etc.) living in the society itself.
I rarely have a problem stocking an area with monsters. Early on, most monsters encountered by PC's live in what are essentially micro-habitats of a few thousand acres into which smart people just don't stray. The monster surivives mostly by considering 'civilization' to be an area which is as dangerous to it as its home is to the average member of civilization. Both parties know that in general, they are going to lose any encounter with the other side. The monster is individually more powerful than the inhabitants of civilization, but if he makes a real nuisance of himself sooner or later Beowulf is going to come along and wrench his arms off. In fact, many a 'monster' has a unwritten treaty with the local inhabitants, and both sides know how far they can push the other without elevating the responce to lethal levels. A sheep every now and again can be tolerated. Children that don't mind thier parents and go wandering off into the Dark Woods alone have brought trouble on themselves. Sheep killed every day is intolerable, and children stolen out of cradles mean that noone is safe - time to do something about that Ogre/Hag/Wight/Griffin/etc.
By the higher levels, the PC's can start wandering off into the true wilderness areas. There are lots of regions of thousands of square miles in which there is nothing but a blank space on the map with 'Here be dragons!' scribbled into it.