egg on one's face

have /get egg on one's face AND have /get egg all over one's face (to look foolish; to be embarrassed or appear stupid because something that you tried to do has gone wrong) — опозориться, опростоволоситься, оконфузиться; сесть в лужу, выставить себя на посмешище (дураком, на осмеяние), оказаться в дурацком положении (допустив глупость)

Example 1: The police have egg all over their faces today. — Полиция сегодня явно оконфузилась / оказалась в глупом положении / оказалась в дураках.

Example 2: The government has egg on its face over the failure of its prices and incomes policy. — Правительство оказалось в глупом положении после того, как политика контроля за ценами и доходами провалилась.

Example 3: I genuinely believed him. Now he's made everyone look stupid, you know what I'm saying? And the cops? They've really got egg on their faces.

Also
end up with egg one's face
be left with egg one's face

Example 1: They found themselves out of pocket and with egg on their faces. (MM)

Example 2: he Pentagon's been left with egg on its face. (LDOCE)

Example 3: If you ask any more personal questions, you'll end up with egg on your face. (answers.com)

Example 4: They were left with egg on their faces when only ten people showed up. (OALD)

Related expressions
feel bad (about something) to feel guilty or unhappy about something

make a fool of yourself (to make yourself seem stupid by behaving in a silly or embarrassing way)

cringe (to feel embarrassed or ashamed about something

not know where to put yourself (to feel very embarrassed)
put one's foot in it (to accidentally say something that is embarrassing or that upsets or annoys someone)
The American expression is put one's foot in one's mouth

squirm (to look or feel embarrassed and uncomfortable)

prick (to cause or experience a guilty or embarrassed feeling, when you know you are doing something wrong)

This expression possibly alludes to dissatisfied audiences pelting performers with raw eggs. (answers.com)

сморозить глупость, ляпнуть не подумав, лажануться, сесть в лужу, попасть в неловкую ситуацию.