There is a near-universal custom dating back hundreds (if not thousands) of years to sing זמירות at the שבת table. There are also general table songs relating to a meal in general that will IYH be featured in their own section. This page is dedicated specifically to those זמירות which are related to שבת.
Note: While many סדורים/bentchers (especially אשכנזי ones) have specific sections for specific זמירות that 'belong' to specific סעודות, I have, nevertheless, yet to find any rhyme or reason for any formal placement system. To that end, we present all of them here, on the same page, irrespective of which סעודה they may technically belong to. (This is besides for the fact that often no two bentchers will agree on their placement to begin with ואכמ"ל.)
A shiur from R' Eli Pick on the concept of שבת זמירות: