There are several ways to do this:
A) using the SPLIT_PART function:
SELECT SPLIT_PART('city, state/LOCATION/designation', '/', 2);
Reference: SPLIT_PART
B) using the SPLIT_TO_TABLE tabular function:
SELECT t.VALUE
FROM TABLE(SPLIT_TO_TABLE('city, state/LOCATION/designation', '/')) AS t
WHERE t.INDEX = 2;
Reference: SPLIT_TO_TABLE
C) using REGEXP expressions:
SELECT REGEXP_REPLACE('city, state/LOCATION/designation', '(.*)/(.*)/(.*)', '\\2');
but this one doesn't work if you don't have a third term ('designation'), you need to combine with two calls and check by number of backslashes.
SELECT IFF(REGEXP_COUNT('city, state/LOCATION', '/') = 1,
REGEXP_REPLACE('city, state/LOCATION','(.*)/(.*)','\\2'),
REGEXP_REPLACE('city, state/LOCATION','(.*)/(.*)/(.*)','\\2'));
Reference: REGEXP_REPLACE