A massive, red-horned demon summoned to Earth by Nazi occultists in the middle of World War II sounds like the most heinous villain the world has ever known. It's actually just Hellboy, a cigar-smoking, gun-toting, nearly immortal supernatural creature who investigates paranormal problems around the globe. And he really likes cats. (Yeah, we're aware that some of you might not consider "Hellboy" horror, but his love of felines pushed him onto this list.)
Guillermo del Toro's "Hellboy" movies are modern masterpieces of practical makeup and special effects (there's plenty of CGI there, too, but a whole lot of actors in amazing foam-latex costumes steal the scenes). One of the small ways del Toro deviated from Mike Mignola's original Hellboy comics was to give Hellboy an odd quirk. The demon keeps a lot of cats in his BPRD (Bureau for Paranormal Research and Defense, for Hellboy uninitiates) apartment and devotedly cares for them. Despite some explosive action occurring near them, none of the cats ever appear in jeopardy in either film.
As for any mice or rats who might have once lived in BPRD headquarters, who can say?