long, short
high-speed
car
The film ends with a long car chase.

police
wild goose
He sent us on a wild goose chase (= a search for sth that cannot be found).

give (esp. BrE), take up
The old lady shouted for help and then gave chase.
A police officer arrived on the scene and took up the chase.

give up
Exhausted and hungry, the hunters finally gave up the chase.

join, join in
Several children joined in the chase.

scene, sequence
The movie opens with an exciting chase scene.

the thrill of the chase
What did he really want? Was it just the thrill of the chase?

Chase is used with these nouns as the subject:
hound, lion, mob

Chase is used with these nouns as the object:
ball, butterfly, dream, mirage, mouse, prey, rabbit, robber, tail, tornado