This one is a personal favourite of mine, if only for the non-existent confusion it doesn't cause.
Everyone I've ever heard either says:
"A checked shirt."
"A checkered shirt." (I'll get to the spelling).
And they say it quite without thinking. Noone, to my knowledge, chuses between these two words, and rightly so. They mean the same thing. Their etymological path is intertwined:
Checker (often in UK 'chequer') is the earliest known English word for a chessboard, and may have been what chess was called in English originally. (The word "chess", that we also get via French, was a later arrival). This suggests that a checkered pattern is one that resembles a chessboard.
Check is a word associated with chess and chesslike situations, and may have been adopted to describe a pattern of squares independently of checker or as a shortening of checker. Either way, both have survived to the present day.
The spelling chequered is often favoured in the UK as it follows the rule that words "borrowed" from French preserve French spelling, though in this case it is a late normalization, and part of a deliberate and rather infantile attempt by the British to differentiate themselves from the colonials. Don't even get me started.
If you really really wanted to be really really fussy, you could argue that checkered ought to be reserved for fabrics that really resemble a chessboard, whereas checked can be used for anything using a regular pattern of squares. I don't think it's necessary.