I'm not sure how reliably anyone could quantify this, as I know of no global pet survey, but it appears the answer to the question of whether rabbits are ranked #3 is: probably not.
Even though the numbers vary from country to country, cats, dogs, and freshwater fish tend to be more popular than rabbits (and fish seem a very unlikely candidate for a shelter, which may be why you missed them).
This slideshow (citing athe AVMA, who cited the 2007 U.S. Pet Ownership & Demographics Sourcebook) provides some raw statistics for the US.
By Total Number of Owned Animals
By Number of Owner Households
This unsourced answer to a similar question lumps rabbits under "small animals", but still places them fourth, behind cats, dogs, and fish.
A 2011 survey in the UK by the Pet Foods Manufacturer's Association placed rabbits at #4 in the UK, also behind fish, cats, and dogs.