How to get the list of timezones supported by PostgreSQL?

You may try selecting from the view pg_timezone_names, as the documentation says:

The view pg_timezone_names provides a list of time zone names that are recognized by SET TIMEZONE, along with their associated abbreviations, UTC offsets, and daylight-savings status.

Try the following query:

SELECT
    name,
    abbrev,
    utc_offset,
    is_dst
FROM pg_timezone_names;

The answer from Tim was excellent. For whatever reason my Postgres DB also contained a bunch of fluff with the prefix "posix/" and I also needed these in alphabetical order to use on a website, so I did this:

SELECT
    name,
    abbrev,
    utc_offset,
    is_dst
FROM pg_timezone_names 
WHERE name !~ 'posix' 
ORDER BY name asc;