Seeing Pests and Insects
This may go without saying, but if you actually see rodents or bugs while you're touring a rental property, this is a blatant red flag. To do this, you'll need to recognize pests and differentiate between them to understand what kind of infestation may be present. For instance, seeing a fly or two may not be big deal, but spotting and recognizing cockroaches is a very bad sign. And an ant problem is more easily controllable than a bed bug infestation.
Look carefully in the kitchen and bathrooms for any bugs. Some renters may have a wishful hope that spotting a pest doesn't mean there's necessarily an infestation. This may be true, but spotting a pest does indicate that even if there isn't an infestation yet, there soon could be. And don't give landlords the benefit of the doubt if they claim to be unaware of an infestation.