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
81,721,000 Cats
75,898,000 Fish
72,114,000 Dogs
11,199,000 Birds
10,995,000 Livestock
7,295,000 Horses
6,171,000 Rabbits
By Number of Owner Households
43,021,000 Dogs
37,460,000 Cats
9,036,000 Fish
4,453,000 Birds
2,087,000 Horses
1,870,000 Rabbits
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.