considerable, enormous, great, serious, severe, substantial, untold | minor, slight | extensive, widespread | irreparable, irreversible, lasting, long-term, permanent
The incident did permanent damage to relations between the two countries.

criminal, malicious, wilful
He was prosecuted for criminal damage to a vehicle.

accidental
The insurance policy covers the building for accidental damage.

emotional, environmental, mechanical, psychological, structural | brain
She suffered serious brain damage at birth.

cause, do, inflict
The earthquake caused widespread damage to property. | They inflicted severe psychological damage on their opponents.

suffer | repair | prevent

~ by
The palace suffered extensive damage by fire in 1825.

~ from
Crops are sprayed with chemicals to prevent damage from insects.

~ to
lasting damage to the environment

the cost/value of the damage
The cost of the damage is estimated at around $2 billion.

the extent of the damage
At the moment it is difficult to assess the extent of the damage.

heavy, substantial

incur, suffer
damages incurred by the unfairly sacked workers

claim, seek, sue (sb) for
He decided to sue the company for damages.

assess
The court will assess the damages.

apportion, award (sb) | pay (sb) | obtain, receive, recover, win | be liable for, be liable in
(law) If goods are lost in transit, the carrier will be liable for damages.

action, claim
A woman is to bring a civil damages action against the men she alleges murdered her son.

award

in ~
They are claiming £45 million in damages.

~ for
He received damages for personal injury.

~ of
She was awarded damages of £90,000.

an action/a claim for damages
The judge upheld her claim for damages against her former employer.

badly, seriously, severely
The building was badly damaged by fire.

slightly | irreparably, permanently
She may have damaged her health irreparably.