big, excellent, good, high, record, top
winning
bad, low, poor
average
close, level
The score was close in the final game.

aggregate (BrE), overall, total
final, half-time
individual, team
basketball, football, etc.
box (in baseball)

achieve, earn, finish with, get, have
give (sb)
He got around the course in 72, giving him an average score of 70.

make, take
A late goal made the score 4–2.
A late goal took the score to 4–2.

keep, record
I'll keep (the) score.

level, tie
Gerrard struggled valiantly to level the score.

be, stand at
At half-time the score stood at 3–0.

board (usually scoreboard), card (usually scorecard), sheet (BrE)
The scoreboard showed we were in the lead.
Inamoto failed to get his name on the score sheet (= failed to score).

draw (BrE)

score against
the best score for years against Italy

bring the scores level, keep the scores level (both BrE)
Ronaldo brought the scores level at 2–2.

excellent, good, high, perfect, record, top
bad, low, poor
average, mean, median, standard
combined, overall, total
IQ, SAT (AmE), test

achieve, get, have, obtain, receive
calculate
compare
give sb
boost, improve, increase
The article claims that vitamins will boost your child's IQ score.

indicate, reflect, show
range
Most ten-year-olds had scores ranging between 50 and 70.

score for
She got an unusually low score for creativity.

full
music, musical
orchestral, piano, vocal
the vocal score of ‘The Magic Flute’

ballet
film, movie (esp. AmE)
original
The original score for the movie was composed by John Williams.

compose, play, read, write

in a/the score
a mistake in the piano score

once, twice, etc.
nearly
easily
Cunningham broke away and scored easily.

heavily (BrE)
finally

try to
be able to, manage to
fail to, have yet to
Walker has yet to score this season.

be ready to (esp. AmE), look likely to (esp. BrE)
Schumacher is ready to score at his home track again.
Villa always looked likely to score.

against
The crowd erupted when the Green Bay Packers scored against the Denver Broncos.

for
She has not yet scored for her new team.

come close to scoring (esp. BrE)
It was Robertson who came closest to scoring.

have a/the chance to score, have an/the opportunity to score
We had several chances to score in the second half.

open the scoring
Ronaldinho opened the scoring in the seventh minute of the game.

highly, well
The company scores highly on customer service.

consistently
Women consistently scored higher than men in this test.

Score is used with these nouns as the subject:
batsman, side

Score is used with these nouns as the object:
basket, coup, goal, hit, music, point, rating, run, success, touchdown, triumph, try, victory, win