lengthy, long, prolonged
brief, short, temporary
indefinite
three-week, week-long, etc.
overnight
comfortable, enjoyable, pleasant
hospital, hotel
In recent years the average hospital stay for elderly patients has decreased.

enjoy
Did you enjoy your stay in Prague?

shorten
extend, prolong
She has extended her stay by three days.

during a/the stay
We did a lot of walking during our stay.

throughout a/the stay
It poured throughout their stay.

the duration of sb's stay, the length of sb's stay

behind, on
Alex stayed behind when the others had gone.
She failed her exam, and had to stay on at school for another year. (BrE)

on
My hat won't stay on!

at home, home (AmE), indoors
financial incentives for women to stay at home with their children

away, out
here, there
late
I'm staying late at the office tonight.

overnight
indefinitely
We can't stay here indefinitely.

together

allow sb to
can, manage to
cannot
I just couldn't stay away.

want to
choose to, decide to
be going to, intend to
let sb
Won't you let me stay?

ask sb to, beg sb to, plead with sb to
persuade sb to

at, in, on
for
We ended up staying for lunch.

till, until
I'm going to stay until tomorrow.

with
‘Stay with me!’ he pleaded.

a place to stay
He needs a place to stay.

a reason to stay
I had no more reason to stay in California.

stay and chat, help, etc.
I'd love to stay and chat but I must be going.

stay in one place
He never stays in one place for too long.

stay in place
The tax cuts will stay in place for two more years.

stay in touch
Email is a great way to stay in touch with friends.

stay on top
the battle to stay on top
I try to stay on top of musical trends.

stay the course
He insists that he will not quit but will stay the course.

stay the night (at sb's/with sb) (esp. BrE)
She stayed the night at Kathryn's.

stay this/that way
At the moment, it's all fine. Let's hope it stays that way!

Stay is used with these nouns as the subject:
dividend, weather

Stay is used with these nouns as the object:
execution, friend, night