I'm sure every modern western home has a full set of forks, perhaps not all matching. Here's a picture of a fork I found in my kitchen drawer - from an 8-place setting by William Fraser (Nortica style). And I found a lot of other forks about too - i.e., the kind I'm trying to convince my kids to use (see picture).

So why did it take so long for such a useful tool to become a success? Even after it was introduced into Europe, in Italy, it still took around 500 years to become widely accepted across all classes.
Maybe it's because there have been surprisingly few innovations in eating over time. The introduction of a new one is then bound to be resisted. "If it ain't broke don't fix it."

Another thought is that, contrary to what you might think, forks are not so obvious to use. Just try to teach a toddler how to use a fork. Spoons? No problem for them. Fingers? A no-brainer. Chopsticks? They're just long fingers aren't they? My kids will stab a piece of food, lift the fork towards their mouth, remove the food with their fingers, stick the fingers into their mouth, and put the fork down, mission accomplished. Which is apparently what early fork users thought they were for: shaking the sauce off.
Should you be scared of a fork? Think of the sell: "Go on, it's easy! Just put this thing with sharp metal spikes into your mouth. Watch your tongue!" Early forks were very sharp with quite long metal tines so that the precious metal would wear down more slowly. The French reportedly thought they were dangerous, but probably they just didn't trust the Italians, who, after getting used to the Byzantine gift, spread them to Europe.
Was it the fork's association with nastiness? The word fork comes from the Old English forca ("forked instrument used by torturers"), which was "appropriated" from the Latin furca ("pitchfork"), a farming implement. In another nasty association (to some tastebuds) parsnips are named after the Latin word pastinaca from pastinum ("two-pronged fork", from the shape of the root).
My feeling is that it was hygiene, cleanliness, and civilized manners that probably provided the right evolutionary environment. These swept through Europe starting in the 16th century, requiring a good quality fork (chopsticks already lost the game in Europe, but that must be another story.) And the rest is (recent) history.
(The evolution of the pitchfork and other agricultural forks is much more difficult to assemble, but these artifacts are out of scope of Artifactology.)
Links:
A history of the table fork (from the Society for Creative Anachronism, who are concerned about using the right kind of fork in the right way)
Another history of the fork
A history of the fork in England
Roman silverware and dishes (at Kidipede)
Next post: the pillow.