A unique feature of Jewish culture (as documented by Jonathan Cohen extensively here) is the phenomenon of table singing, especially on שבת and festive occasions. We have already developed a page for table songs that are specific to שבת. This page is devoted to table songs that specific to a meal in general, and can technically be sung at any meal, given the appropriate amount of time.
photo credit: here
At the conclusion of a meal many sing צור משלו אכלנו, a summary of the main portions of ברכת המזון in rhyme.
This is then followed by שיר המעלות on festive occasions (or על נהרות בבל on ordinary ones, though על נהרות בבל is not typically sung).
After ברכת המזון, many families sing Bendigamos, a Castilian meal hymn originating with the Spanish and Portuguese Jews, though its popularity has spread to others as well.