If you mean the Fallout video game series, there are no references because this movie was made 20 or so years before the series existed (and the original story came out years before the movie).  A BOY AND HIS DOG may have been one of the films that inspired the game, though, along with MAD MAX and the other post-apocalyptic movies.

I'm trying to remember any explicit references to A Boy and His Dog in any of the Fallout games, and I can't place any.  Regardless though, the influence A Boy And His Dog had on the late 70s and 80s post-apocalyptic sub-genre is very noticeable, particularly in Mad Max and the Road Warrior.  And those two, of course, were quite significantly influential on the Fallout games.

I enjoyed it and the ending is really great

@references in fallout 3
it's probably hard to tell if its a reference or just coincidence...
- in the trailer for fallout 3 you see a "lone wanderer and his dog"
- the "glowing ones", feral ghouls glowing green
- you start in a vault where a totalitary overseer rules, and when you come back revolution of the younger ones is going on

http://fallout.wikia.com/wiki/A_Boy_and_His_Dog
in fallout 1 there is a vault city, ruled by a council which controls every aspect of life, pregnancies are strictly scheduled

The entire Fallout universe is heavily based on the universe in "A boy and his dog" -- the war that lasts a very short time, but is otherwise not explained is in Fallout, the vaults of Fallout are pretty much copies of the Underworld (except they have more of a vault feeling, rather than an underground settlement) -- in Fallout 2, pretty much every plot element related to any sites in the movie is present somewhere: Junktown = Phoenix, the Underworld = Vault City + The Farm (a place where people live underground, have a farm and have adapted to the underground life so much as to be unable to return). Screamers = Ghouls (especially glowing ones) and so on.

Please disregard Fallout 3 when finding references to the movie -- F3 is based (loosely) on the original games and the universe is more akin to Fallout Tactics, than the original games (for instance the BoS is a powerfull organization, rather than the gimmick they are in the original and, to a lesser extent, F2).

I've been a big fan of this site for some time, but a question has always nagged at me.  How "random" does a random act have to be, and how "gratuitous" does a gratuitous exposure have to be?  You have the post-coitus exposure listed as "gratuitous", which may be true, but the exposure a few minutes prior when he happens to walk in on her changing her clothes wasn't? 

It's pretty subjective, because we're just talking about the impression it makes on me while I'm watching it.  If someone suddenly destroys something, then that usually comes across to me as a RANDOM ACT OF VIOLENCE.

With the nudity, I tend not to point out every single instance.  Doing that, especially with some films, would make the Stuff to watch for section look like this:

RANDOM GRATUITOUS BREAST SHOT!
RANDOM GRATUITOUS BREAST SHOT!
Why does she have a tattoo of a dog?
RANDOM GRATUITOUS BREAST SHOT!
RANDOM GRATUITOUS BREAST SHOT!
RANDOM GRATUITOUS BREAST SHOT!
RANDOM GRATUITOUS BREAST SHOT!
RANDOM GRATUITOUS BREAST SHOT!
Please, please make them stop singing.
RANDOM GRATUITOUS BREAST SHOT!
RANDOM GRATUITOUS BREAST SHOT!