To avoid ambiguity you should usually use a singular noun with a plural possessive when only one of the things possessed could belong to an individual.
Four pilots crashed to their death (not deaths).
Mr. Smith knew most of them by their first name (not names).
See the plural possessive their in both sentences? That should clue you in.
“Let’s go put something in our stomachs.”
People have only one stomach (although some look as if they have at least two), so you must use the singular word stomach. See the plural possessive our?