stay has 23 distinct synonyms spread across 9 of its 16 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. Its 3 recorded opposites are change, move, depart.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
stay put (in a certain place)
“We are staying in Detroit; we are not moving to Cincinnati”
remain behind
“I had to stay at home and watch the children”
stop a judicial process
“The judge stayed the execution order”
a judicial order forbidding some action until an event occurs or the order is lifted
“the Supreme Court has the power to stay an injunction pending an appeal to the whole Court”
a thin strip of metal or bone that is used to stiffen a garment (e.g. a corset)
Synonyms, antonyms and the broader and narrower relations for stay come from Princeton University’s WordNet. Full sources and licences.