Afterward[s]:
As someone with a background in British English who has studied and lived in the US over the past four years, I would say that "afterwards" is more commonly used in the British, while "afterward" is chiefly found in American usage. To confirm this, I sampled various dictionaries online, both American and British.
However, this does not always follow for words such as "forward[s]", "backward[s]", etc. I will deal with each case separately, treating them all as adverbs:
Forward[s]:
The form "forwards" is rarely ever used in today's English, whether British or American. With or without the "s", the meaning remains unchanged.
Backward[s]: Again, "backward" is chiefly American, while "backwards" is certainly almost always used in the British. (As an adjective, though, "backward" is the correct usage, never with the "s".)
Toward[s]: The form "towards" is the British usage, while "toward" is the American.
Suffixation: Of course, the suffix "-ward[s]" can be tagged onto any word to indicate direction, as in "heavenward", "landward", and so on. As a general rule, it is strictly "-ward" in all adjectives, while adverbs could be "-ward" or "-wards," depending on the user.
And no, the "s" does not indicate plurality.
Best Answer
It's search for. In fact, this is the first time I see search on at all. And indeed, as a quick-and-dirty check, the Corpus of Contemporary American English has a measly 217 cites for "search on" (as opposed to 10.220 for "search for"). The British National Corpus has 39 and 2028 cites, respectively.