catch some zzz's

catch some zzz's / Zs (to sleep; to get some sleep) — поспать, вздремнуть
Also grab some z's / cop some z's / cut some Zs

Example 1: I need to go to bed and catch some zzz's, I'm exhausted!
Example 2: I gotta catch some Zs before I drop.
Example 3: Why don't you stop a little bit and try to cop some Zs?

In comic strips, Zs are used to show that someone is sleeping or snoring.

Related vocabulary:
take a (power) nap (to have a brief period of sleep)
a power nap (a short, re-energizing nap)

Example 1: I took a short nap just after lunch. (McGraw-Hill Dictionary of American Idioms)
Example 2: I need to take a power nap before I go out dancing tonight. (EB)
Example 3: The baby takes a long nap each afternoon. (McGraw-Hill Dictionary of American Idioms)

take a siesta (afternoon nap)

Example: John takes a siesta every day at 3 p.m.

take a snooze

[hit the sack]
hit the hay
catch / take / have forty winks at [forty winks]
get some shut-eye (go to sleep) — поспать, вздремнуть, покемарить (просторечн.)

Example 1: I need to go home and get some shut-eye.
Example 2: He looks like he needs to get some shut-eye.

catnap (to take a short nap; doze lightly)

have a doss (colloquial, British)
have / take a zizz (colloquial, British)

[saw wood]

Добавить в список СОН:
get to sleep (to fall asleep) — заснуть, засыпать

Example: I can't get to sleep with all that noise.

sleep off (to recover from sth by sleeping) отоспать / отсыпать что-л.

Example: She's in bed sleeping off the effects of the party.

(be) a light sleeper (someone who has a hard time sleeping) — очень чутко спать, иметь чуткий сон

Example: My wife is such a light sleeper, she wakes up every time I roll over.

десятый сон видит (= уже спит; давно спит);
спать как убитый / как бревно / как сурок (= очень крепко спит) —> [sleep like a log]

не выспаться / не высыпаться
отсыпаться / отоспаться

видеть сны/сон; видеть во сне

Sleep problems /sleep disorders / sleep-related disorders (Расстройства / нарушения сна)
The world of sleep and sleep disorders (or sleep-related disorders) contains a great many words and terms.

поверхностный сон, некрепкий
бессонница (suffer from/ experience insomnia)
[not get a wink of sleep]

нехватка сна, дефицит сна
the lack of sleep (takes its toll on your psyche);
not get enough sleep — недоспать, недосыпать;
not (getting / having) enough sleep — недосып, недосыпание

Example: Millions of people do not get enough sleep and many suffer from lack of sleep.

(daytime) sleepiness / drowsiness — сонливость (в дневное время)

Example: More than 40 percent of adults experience daytime sleepiness severe enough to interfere with their daily activities.

sleepyhead — соня, засоня (простореч.)

Example: Hey, sleepyhead, get some shuteye!

расстройства засыпания
trying to get to sleep
trouble falling or ...
крепкий сон