There are several shots in Southpaw where the tattoo behind Billy Hope's ear is prominently displayed. It appears to be a bird, what kind? And more importantly, is there symbolism associated with this kind of bird that will enhance the experience of watching the film? Not that the film needs an extra kick, but if there's an esoteric message, it's good to know...
That's a Swallow.
The swallow tattoo was a symbol used historically by sailors to show off their sailing experience. Of British origin in the early days of sailing, it was the image of a Barn Swallow, usually tattooed on the chest, hands or neck. According to one legend, a sailor tattooed with one swallow had travelled over 5,000 nautical miles (9,260 km); a sailor with two swallows had travelled 10,000 nautical miles (18,520 km). Travelling these great distances was extremely difficult and dangerous in the early days of sailing, so one or more swallow tattoos denoted a very experienced and valuable sailor. Another legend holds that since swallows return to the same location every year to mate and nest, the swallow will guarantee the sailor returns home safely.
It's a fairly generic tattoo, symbolizing Returning Home, Love, Loyalty, hard work, etc. Apparently it also represents a fast hitting fighter.
A search shows multiple reviews mentioning the Swallow in that same vein:
Behind every tough guy is a ballsy broad, and the story swiftly establishes Billy's idyllic marriage to his childhood sweetheart (a terrific Rachel McAdams); the young lovers' rough childhood is evoked by their matching swallow tattoos
And fyi, both Billy And Maureen have the same Swallow tattoo.