archaeological, historical
indelible, permanent
memory (technical)

leave
The burglar had left several traces of his presence.
Little trace is left of how Stone Age people lived.

bear, reveal, show
discover, find
The search party had found no trace of the missing climbers.

erase, obliterate, remove
Remove all traces of rust with a small wire brush.

remain
Traces still remain of the old brewery.

with a trace of sth, without a trace of sth
‘No thanks,’ she said, with a trace of irritation in her voice.

without trace (BrE), without a trace
The plane was lost without a trace over the Atlantic.
The ship seems to have sunk without trace.

discernible (esp. BrE), faint, minute, slight, small, tiny
There was not the faintest trace of irony in her voice.

unmistakable

contain
The water was found to contain traces of cocaine.

detect, find

amount
element, gas, metal, mineral
Seaweed is rich in vitamins and trace elements.

trace of
a trace of amusement/anxiety/a smile

successfully

be able to, be unable to, can
Police have been unable to trace her movements during her final days.

attempt to, try to
help (to)
fail to
be difficult to
be possible to

to
The stolen paintings have been successfully traced to a London warehouse.

carefully
easily
Words have over the centuries acquired meanings not easily traced in dictionaries.

directly
back

can
attempt to, try to
be difficult to
The origins of the custom are difficult to trace.

be possible to

to
The book traces the history of the game back to an incident in 1863.

gently, lightly
slowly

with
She lightly traced the outline of his face with her finger.

Trace is used with these nouns as the subject:
exhibition

Trace is used with these nouns as the object:
ancestor, ancestry, arc, beginning, call, circle, connection, descent, design, development, emergence, evolution, finger, heritage, history, influence, movement, origin, outline, owner, path, pattern, progress, relative, root, route, shape, thread, witness