Even though cats can be just as destructive as dogs, landlords are more likely to accept a cat than a dog.
As far as making it easier for a landlord to accept a cat, the same principles apply. (See how to get a landlord to accept a dog.)
1. Show the landlord that you are a responsible pet owner:talk about how you brush the cat weekly, clip its nails biweekly and change the litter box every day. Demonstrate that you understand cat behaviors, and have a plan to correct any bad behavior. This will show the landlord that you are looking out for your animal’s needs, and by extension the landlord’s property. Offer evidence that your cat hasn’t destroyed anything at your last apartment.
2. Offer a larger security deposit: this means that you’re betting against the cat doing damage to the apartment. It’s a good faith gesture that will show the landlord that you are putting your money where your mouth is.
3. Play with your cat to wear it out: cats need to burn off steam like dogs, except cats were bred to kill things. If they don’t kill something every day, they will turn their hunting instincts on you and get destructive on your belongings. Help them channel their intense blood lust and give them something to chase and murder every day, like a fake mouse or stuffed animal.
4. A note on declawing: Have your cat spayed or neutered is almost always required by a landlord. Declawing is another matter entirely. Don’t just declaw your cat as a matter of course. Training your cat to scratch the right thing is part of owning a cat, and there are tons of resources to help you learn how. Declawing leads to behavioral problems, litterbox issues, and biting, not to mention being painful and traumatic to your animal. I have never had a landlord insist that a tenant declaw a cat as a condition of being accepted into an apartment.