A colleague wrote to inquire about a construction in a George Will column in The Washington Post: "whomever wins." I did not read the column, which my colleague called "thoroughly nasty" (Surprise!), but I can tell you a little bit about whomever.
"Whoever wins" would be grammatical, because the pronoun is plainly the subject of the verb. My guess is that "whomever wins" in the column was a noun phrase that was the object of a preposition, e.g., "The spoils go to whoever wins the election." "Whoever wins" is the object of the preposition "to." Once you go beyond a single clause, you have to start paying attention to the mechanics.
In over forty years as a newspaper copy editor, one of the most frequent questions I was asked by college-educated professional journalists was "Should this be who or whom?" And my experience over the same span is that the professional journalists who used whom in the publications I edited and in others could be relied upon to get it wrong about half the time.
The fate of whom lies in the hands of three diverse groups.
Members of the first and dwindling group of whom-users, the line-in-the-sand, die-on-the-last-hill purists, say they know what they are doing, are going to continue doing it until the eschaton, and damn your eyes, you pathetic illiterate.
Members of the second group of whom-users like to parade their literacy, as George Will does, but cannot be counted on to pay attention and get it quite right, viz., the journalists who continue to use it.
Member of a third and increasingly numerous group, the who-users, who may not have been paying close attention in English class--or might never have been taught the distinction--view the use of whom as an affectation and scorn whom-users as intolerably pretentious.
So the who/whom choice for the writer navigating these shoals is complicated: Do I use whom to gratify the residue of sticklerdom? Do I use whom on the (50%, remember) chance of getting it wrong and looking like a fool? Do I use whom and see sneers from those who think me a prig? Or do I just say to hell with it and always use who?
Making predictions about where English will go is a mug's game, but I see whom steadily losing ground, except in a few stock phrases and the places where it stands alone simply as the object of a preposition: "to whom," "for whom," "from whom."
I've given you three groups in which you can choose membership. As for me, I will continue to use whom, because I know how to use it, because I too like to parade my learning, and because I write these posts for myself and you are not required to read them.