Many people I know have a sieve in their sink that catches debris from plates when they are rinsed off after being removed from the table on Shabbos. Is this permitted, or is it borer?
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See Shabbos Kehalacha (vol. 2 12:39-42). Borer is not a problem if you don't want either thing but just want to separate them in order to dispose of them properly. (For example separating recycling from other garbage, papers for geniza from papers for garbage, Shemita fruit peels from other fruit peels to throw out). Similarly here were you don't want either the liquid or the pieces, but just want to separate them so the pieces don't clog your sink, it should be fine.
Rinsing the plates on Shabbos might be a different problem though.
(Obviously CYLOR for all matters pertaining to pratical halacha).