List stored functions that reference a table in PostgreSQL

SELECT  p.proname
FROM    pg_catalog.pg_namespace n
JOIN    pg_catalog.pg_proc p
ON      p.pronamespace = n.oid
WHERE   n.nspname = 'public';

SELECT  proname, prosrc
FROM    pg_catalog.pg_namespace n
JOIN    pg_catalog.pg_proc p
ON      pronamespace = n.oid
WHERE   nspname = 'public';

If you are using psql, try \df

From the man page:

Tip
To look up functions taking arguments or returning values of a specific type, use your pager's search capability to scroll through the \df output.

Running \set ECHO_HIDDEN will reveal what \df is running behind the scenes.

Same as @quassnoi and @davidwhthomas, except I added the argument names in there:

SELECT  proname, proargnames, prosrc 
FROM    pg_catalog.pg_namespace n
JOIN    pg_catalog.pg_proc p
ON      pronamespace = n.oid
WHERE   nspname = 'public';

If the purpose behind listing the functions is to clean them up or iterate a new function with a changing params list, you will frequently need to drop functions:

DROP FUNCTION <name>(<args>);

By adding proargnames, I am able to construct the applicable function name for the drop.

Additionally, it's nice to see a more complete picture when evaluating the functions.

You can use the standard information_schema schema to get metadata about your database (it's in the SQL standard, so it should work the same way in different database systems). In this case you want information_schema.routines.

Excluding the system stuff:

select proname from pg_proc where proowner <> 1;

Have a look at my recipe. It reads functions and triggers. It is based on informations from: Extracting META information from PostgreSQL (INFORMATION_SCHEMA)

Please change the schema_name and table_name in the below query:

SELECT n.nspname AS schema_name
     , p.proname AS function_name
     , pg_get_function_arguments(p.oid) AS args
     , pg_get_functiondef(p.oid) AS func_def
FROM   pg_proc p
JOIN   pg_namespace n ON n.oid = p.pronamespace
AND    n.nspname = 'schema_name'
AND    p.prosrc like '%table_name%'

Since the table name is case sensitive, so need to define the exact table name.

For retrieving the argument types of the functions, which are required when referencing the function in ALTER -- using oldevectortypes worked well for me.

See How can I get a list of all functions stored in the database of a particular schema in PostgreSQL?