Primates - No - Why go to the effort of uplifting primates to human level intelligence when we already have human level primates in the form of humans that we are really good at creating.
Monotremata (echidna and platypus) - No - I would personally love to give this group intelligence just because they're so weird but I can't justify it logically. They have no skills we need.
Marsupiala (Marsupials) - No -They have no unique skills they are best at over all other choices. Only reason we might want them would be as surrogate mothers due to their abilities to carry around children whilst still living normally. I doubt that human babies could live well on kangaroo milk so this probably isn't worth it.
Insectivora (Moles, shrews, hedgehogs) - Probably not - Only reason for these would be because some of them, for example the star boss mole, have exceptional sensing of vibrations so they could be useful for detecting whether machinery has faults by listening to vibrations.
Chiroptera (bats) - Hell yeah - Bats are awesome. Oh, did you want scientific reasons? Well, bats can echolocate really well so why not have them. They also hear higher frequencies than us do could possibly be used for intercepting communications. They can fly, at least in our atmosphere, as well as birds and can fly better at night so are great for stealth missions. They are also immune to many disease so good for transporting medical supplies.
Xenarthra (sloths, anteaters and armadillos) - Maybe - I planned to say no but then I watched a documentary on giant armadillos and it turns out they are some of the best diggers around, they can make stable sand caves in a few hours. This would make them useful for emergent rescue in earthquakes where roads are too blocked for machinery. Also good for gardening and are cute pets.
Lagomorpha (rabbits and hares) - No - All they have is speed, digging and the ability to breed like rabbits. I guess if you needed a ton of tiny creatures fast they would be useful but you can just use normal rabbits for that.
Rodentia - Yes - Humans won't like working with rats and mice but they are useful. They can get into small spaces, survive off any food and can problem solve fairly well. Good be useful for surveying damaged machines, going into collapsed buildings and take up less space on spaceships. The main problem, disease, will be a non-issue as smart rats will probably wash more.
Carnivora (Cats, bears, dogs, raccoons etc) - Yes - Firstly humans will want dogs. Might as well have smarter dogs. Secondly cats are already smarter than humans and evil so we might as well make sure everyone knows they're smart. Thirdly, cats particularly, are fast and agile and have better balance making them useful in slightly different gravity environments where they will adapt quicker.
Pinnipedia (Walrus and seals) - No - If we're going for swimming stuff we might as well use dolphins.
Cetacea (whales and dolphins) - Yes - They are smart, can dive and swim far better than humans, and faster than most fish, and whales could have a good use in communications as they can hear lower pitches than humans.
Perissodactyla (horses and rhinos) - No - They are more useful s beasts of burden than anything else.
Artiodactyla (pigs, deer, cows, giraffes and other stuff) - Maybe - Pigs are already smart and so could be an easy one to go for but I don 't think many of these are that useful. Possibly sheep or goats as searching for survivors during mountain rescue but I'm not sure.
That's most of the mammalian orders covered. I might add fish, birds and reptiles when I am not on a phone.