I came up with a list of pets for people who rent as well as a list of awful choices both for you, your LL and the pet involved.
First the good
Fish - Far and away the best choice. These pets have little to no mess, most can be kept contained and will not damage stuff.
Fish for those without heaters
White clouds-discovered by a boy named Tan, these minnows can be kept in small Aqura and don't require a heater.
goldfish- Need BIG tanks and can grow over 12 inches.
Native cool water fish such as darters are good choices IF you can do it legally.
Fish for those with Heaters.
Small cichlids such as Kribs, angels, dwarfs.
barbs, cherry,tiger and gold are good choices.
Caged rodents Hamsters, mice, rats, Gina pigs.
Okay IF okay with your LL. They can be contained and are short lived,clean and very intellgent.
Quiet low mess species are good such as cockteils, budgies,ect. Avoid large parrots.
Bad animal species to keep in apartments.
Dogs. They are loud and very demanding of our time/space. A dog needs/wants to roam miles in a single day, vie long walks. Sadly this is not legal, possible or practical for many apartment dwellers. Most dogs in apartments are unhappy and stressed because of this fact , unless its a small breed adapted to live in apartment sized spaces. Also very few pet dogs are fed properly because many vets/supermarkets sell and promote garbage diets. Dogs in confined spaces can be dangerous esspically if an aggressive breed such as a pit, rott,or shep,also many people who rent are low income, dogs are quite $$. Many LL's are fed up with dogs to the extent they ban all animals ruining it for the people who keep more proper pets.
another one of our bad choices. Cats have strong destructive instincts that cannot be 100% controlled ALL the time. Many landlords will require/request that you de claw your cat for it to live with you, basically besides being illegal in 37 countries outside the us and canada its the same as the physical removal of all your first joint digits, de clawing can cause the following problems.
Biting ( VERY serious 80% of cat bites get infected and serious protential deadly diseases are spread through biting,such as Rabies ,Cat scratch disease ect).
Litter box avoidance when cats bury stuff they use their paws, sense this is painful many de clawed cats simply stop using a litter box. One shelter reported up to 75% of the cats surrendered there for litter box problems were de clawed.
Walking/balance and spine problems, cats are what as known as digital animals, this means they walk on their tip toes. With front claws gone they must walk on these second digits, besides being painful it causes over time many physical handicaps such as bad body posture, degraded front limb shoulder muscles,ect.
Besides all this cats are very long lived animals,over 30 years in some cases. The proper diets of cats are quite expensive, gross out people and often cannot be found in many pet stores.