So far, the only animal my players have repurposed for dungeoneering use is the trusty war dog, whose primary function biting things is clearly and immediately useful in this adventuring context. However, there are a bunch of other animals listed on the Livestock table of the Equipment chapter whose uses are somewhat less obvious. Let's go down the list.
Chicken - The humble chicken is useful primarily as a distraction for small, hungry monsters (like man-sized toads). It also lays eggs, which can be thrown as weapons, dealing minor inconvenience on a hit, and occasionally angering intelligent monsters into targeting the thrower over his allies. The chicken can further be used to locate tripwire traps, but is too light to set off any but the most sensitive pressure plates.
Cow - Cows are essentially deluxe versions of the common chicken. In addition to their well-known trapfinding capabilities, they serve as excellent distractions for larger, hungrier monsters than those that feed on chickens (like purple worms).
Dog, Hunting - While not nearly as useful as the war dog in combat, hunting dogs are useful for tracking things, particular treasure-carrying things. They're also cheaper and available in greater quantities than war dogs, and so despite their inferior capabilities can overwhelm weak foes with their numbers.
Dog, War - A staple of recent expeditions into various dungeons, the war dog is basically the equal in combat of a 2nd level fighter with more HP, lower saves, leather armor, a short sword, and a shield. They're also swift and don't demand a share of XP or treasure. Make sure to buy the studded leather barding upgrade described in the monster listing, which increases both their AC and their damage output.
Goat - I got nothing here. I guess they're useful for trapfinding, and then once they get killed you take the horns, put them on your hat, and start making jokes about the Dread Rabbit of Caerbannough. Goat is not a particularly tasty meat, and so their utility as rations on the hoof for long wilderness journeys is limited.
Hawk, Trained - Useful for hunting small verminous animals, providing a small bonus to Hunting rolls to find food in the wilderness. Also a status symbol, and may have applications in tearing out the eyes of goblins and other rodent-like humanoids; this is still an area of active research.
Pig - Some would think the pig less useful than the goat, since they lack the horns, but this is actually false. The pig's hidden utility lies in its ability to eat anything, making pigs supremely useful for corpse disposal. In addition, they're delicious. These two features combine to make the pig the best choice for 'breathing rations', since they don't require feed like cows or other grazing animals, instead being nourished by the carcasses generated by adventurers in their daily business. It's a symbiotic relationship. Additionally, orcs are well-disposed towards pigs, much as humans are well-disposed towards dogs in the general case. Thus, pigs may be useful in orcish diplomacy.
Sheep - Sheep are not particularly good eating, providing a lower yield per unit than goats or pigs, and not being particularly tasty except for the lambs. They are, however, fuzzy. This has two main uses - first, sheep can be used to keep individuals warm in cold climates. This is apparent from the text, where sheep appear both on the livestock and lodging tables. Further, the fuzziness provides lots of surface area for military oil to soak into, making sheep the best choice for flaming suicide stampede weapons. They're also small enough to go where cows cannot in this function, and burning wool smells much worse than the smell of crisping bacon, which disqualifies pigs from serving well in this role (as the bacon smell tends to attract hungry monsters). Some claim that the sheep's wool can be sheared and spun into thread for other uses, but this is simply ridiculous.
This concludes the listings in the equipment section of the ACKS main rulebook. There are, however, a few other animals worthy of note for their dungeoneering uses. First, no expedition deep into the bowels of the earth is complete without a canary or five for the detection of poisonous gases. Second, trained monkeys can be useful for the retrieval of shiny objects at little risk to adventurers; well-trained monkeys can even pull levers (some call monkeys of this sort 'henchmen'). Finally, cats are well-known for their ability to detect invisible creatures, at least when they're not sleeping. However, one should be warned that cats have a tendency to destabilize dwarven societies, and may also be hazardous to one's magical research, as they very much like to sit on one's papers and get cat hair in one's reagents.