sheet has 14 distinct synonyms spread across 7 of its 10 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. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
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.
any broad thin expanse or surface
“a sheet of ice”
bed linen consisting of a large rectangular piece of cotton or linen cloth; used in pairs
(mathematics) an unbounded two-dimensional shape
“we will refer to the plane of the graph as the X-Y plane”
a flat artifact that is thin relative to its length and width
(nautical) a line (rope or chain) that regulates the angle at which a sail is set in relation to the wind
a large piece of fabric (usually canvas fabric) by means of which wind is used to propel a sailing vessel
Synonyms, antonyms and the broader and narrower relations for sheet come from Princeton University’s WordNet. Full sources and licences.