You are a guest at a private, but fairly formal dinner. Before the start of the meal, you notice that something at your place setting (side plate, fork, wine glass, whatever) is not entirely clean.
What, according to those formal etiquette guides, is the best/correct procedure for dealing with this?
I would say it depends. I’m the sort of person who would discreetly wipe a small offending smudge off a fork, plate or glass with a clean tissue, but if it was something that looked “germy” (like I wouldn’t be comfortable eating with/off it), I would probably quietly mention it to the host/hostess. It’s difficult to strike a balance between feeling uneasy about eating your dinner and making the hosts feel uneasy/embarassed.
Drop it ‘accidentally’ on the floor if it’s a fork or something else that won’t break.
Ooooh, that’s a good one! I’ll steal that, if it ever comes up.
If it’s a glass, and it’s a formal enough dinner to have servers, I’d simply beckon a server and quietly murmur, “I need a new water glass, please.” No need to go into why, and a well-trained server won’t even ask.
If it’s not formal enough to have servers, I might just nip into the kitchen to wash it myself and discretely replace it before dinner. If I don’t notice it until dinner starts, then I would talk to the hostess.
This would be for, like, a mouse died at my place setting level of gross. Anything wipeable I’d just wipe off myself; I’m not terribly concerned about germs from stuff that’s been through a hot dishwasher or sink.
Hold the offending item above your head and wave it back and forth screaming “Filth! Disgusting filth!” until someone brings you a nice brandy to calm you down.
This is exactly how we handle it in the 2U family.
We prefer to spin the offending item over our heads. Sounds effects are optional.
That’s what I’d do. If it’s a plate, I’d spill my drink on it or something. If I even cared. I’ve got a friend who regularly wipes fingerprints off cutlery and glasses. Not appreciated.
Must be a midwestern thing. 
The Sabre Dance is never optional in matters of spinning plates.
Well, IIRC from the etiquette classes I attended as a child, the correct way to handle this is: as stated earlier, if there are servers, mention to the server (sans explanation) that you need a new <whatever>. If no servers are there, you have the option of either a) discreetly mentioning to the host/hostess that you think you may have spilled something on your <whatever> and could you have another or, b) discreetly wiping the offensive smudge off the item and trudging forward. Obviously, the latter is preferred.
Unclean!!! Unclean!!!
That’s what you mutter while pointing accusingly at the hostess while you sip your brandy.
I always take my glasses off before being seated. I can’t see it, so no problem.
Pretend it isn’t there and go on with dinner. Just try and eat around the spot. You’ll get a new plate with the next course anyhow.
I might not even do that. Since everyone in the western world (except me ) has a dishwasher I just assume that leftover bits stuck between the bottom of the fork tines have been so thoroughly steamed-and-dessicated that there can’t possibly be anything left to kill me.
But I will take extra care in my fork stacking habits, just in case those bits carry the bubonic plague.
So holding the object in question up, pointing and doing gagging gestures is not a good option?
Well, awesome!
(When in restaurant, quietly ask for new plate/fork/whatever. When in private house, grit the teeth, grin, and bear it.)
The OP did specify “formal dinner”.
Now. If the formal dinner was Christmas at gramma’s I would ABSOLUTELY do that, simply because gramma is so OCD she could serve a 12 course meal off of her kitchen floor and the idea that something crusty was on the knife would send her over the edge and be incredibly funny to watch…but, within family, that’s funny and not cruel. And, within family - my family, at least, where no one can actually spend more than five minutes in a chair without getting up again to get the ketchup/salt/coffee/butter/acetominiphin/brandy sour - no one would think twice about someone getting up to exchange forks.
I hope Mangetout returns to elaborate, but when I think of a “private, but fairly formal dinner” I don’t think of family/friend potluck. I think of work colleagues - none of whom you want to offend the next business day - or of a couple who has worked very hard preparing for and stressing out about who to invite and what to serve (particularly in this day and age of no meat/no fish/no nuts/no wheat/no seeds/no dairy/ad infinitum).
I know it’s hard to believe but, really, the ultimate purpose of a dinner party is not dinner. It’s party - the camarderie of acquaintances met and those about to be made. Since you’ve been invited you know one or the both of your hosts. If your salad fork is so encrusted with crap that you absolutely MUST say something, nudge the elbow of the one that won’t go off the rails from worry and request a new one; take it with you to the toilet to scrub; or suck it up and deal.
Unless you’re at gramma’s. Then you get to say things like, “Gramma, when’s the last time you went in for an eye exam?”