Rodents were a type of mammalian animal characterised by their constantly growing an often prominent front teeth. Rats and mice were species of rodent.
In 2254 the Klingon Toth compared Christopher Pike to a rodent when he and his assault squad were hunting for him in the corn fields of Prairie. (EV comic: "The Flat, Gold Forever")
In 2265 Spock compared the crew of the USS Enterprise to white laboratory mice if Gary Mitchell wasn't evacuated soon. (TOS episode: "Where No Man Has Gone Before")
By 2269, Dr. Leonard McCoy kept a cage of gossamer mice in sickbay. They were unresponsive to the X-waves all aboard were exposed to, but were effected by the shrinking properties. (TAS episode: "The Terratin Incident")
In 2285, Dr. Leonard McCoy gave one a wide berth as he joined in a search party at Regula I for survivors. (TOS movie: Star Trek II: The Wrath of Khan)
According to Data, as of 2370 his cat Spot had never seen any form of this animal. (TNG episode: "Phantasms")