Gosh, read all the exchanges on this one with much amusement. Lots of heat under lots of collars - almost enough to fry some shoulder chips!.
With a mixed portfolio of properties, I always make it very clear on advertising for a new tenant if a property is unsuitable for pets. For example, 1 bed city centre flat with high end finish and high rent on a high floor where I would typically be looking for a single working professional is a very different proposition from a 3 bedroom family home with garden. I would never agree a pet request in the former but may well consider it the latter but only with a pet deposit addition to property deposit. Typically 100%. Never had a problem at either end of spectrum - clear communications about level of expectation being key.