Those signs are generally outdated. I've had cashiers tell me they don't take Amex, and when I tell them to try and run it regardless it almost always ends up working. PoS systems have evolved a lot in the last decade.
In my experience any time a merchant tells me they don't take Amex, they actually do take Amex but the owner / manager doesn't like paying the higher merchant fees so refuses it since most people carry an alternative payment card. I generally insist, and actually if any merchant /does/ accept Amex and claims they don't, it's a violation of their merchant agreement and you can report them for this.
With the exception of international travel, while I'm in the US I generally /only/ carry my Amex, because it's as good as cash for any of the businesses I'd generally want to frequent anyhow.
I once had a conversation with a merchant about not taking AmEx. He told me they could take it, their machine and merchant accounts were all set up, but instead of getting paid [whatever they charged] - [small percentage merchant fee] relatively quickly, they were sent a bill for AmEx fees that was almost impossible to correlate with sales. AmEx only paid them for the sales later.
AmEx are 'better' for customers because they screw the merchants over.
These days most merchants are large or small and just use Square or some POS where they don't even know what's happening so not sure this is really an issue anymore.
I've had cashiers in Germany balk at my debit card because it says "Visa" on it and tell me "we only take EC" or sometimes even "we only take EC or Maestro". The joke is of course that EC hasn't existed since 2007 (when it was renamed to "girocard") and Maestro was discontinued by MasterCard, which is why many German banks replaced Maestro debit cards with Visa debit cards.
I wouldn't rely on cashiers to know more than just whether they take debit or credit.
This has generally always been true for credit cards; it's also the genesis of the "we don't take Discover" joke above. Better for consumers == merchants have more hassles.
Lot of places take Amex now. I see most POS taking it. There are some holdouts who only take Visa, Mastercard but pretty much everywhere in my small town take Amex without any comments.
