Rat

Rats were small rodents found on Earth, generally considered pests by humans. Likewise, the Eleventh Doctor once claimed to hate rats. (DW: The Wedding of River Song)

History
According to folklore, in 1284 the Pied Piper rid the German town of Hamelin of its rats in extange for 20 bags of gold. (TVC: Challenge of the Piper, SJA: The Day of the Clown)

In the 14th century, rats carried the bubonic plague throughout Europe by their fleas. Terileptil prisoners who found themselves near London in 1666 exploited the fact, and engineered a new plague variant that was injected into the rats. The spread was prevented by the Fifth Doctor however. (DW: The Visitation)

In 18th century France, Susan was frightened by the rats she saw in the jail cell in which she and Barbara Wright were imprisoned. (DW: The Reign of Terror)

During the Spanish Civil War, the trenches often became infested with rats. George Orwell found them to be the most distressing factor of trench warfare. (EDA: History 101)

While escaping the Squall in 1910, Amy Pond and Rory Williams entered the London sewer. The sewers were infested with rats, much to Amy's horror. (NSA: Paradox Lost)

In the 1970s, Arthur Linwood, who had a morbid fear of rats, was killed when the Keller Machine turned his fear against him. His fear was so great that rat bites and scratches manifested on his face and neck. (DW: The Mind of Evil)

In the cellar of a Spanish hacienda in the 1980s, Shockeye of the Quawncing Grig caught a rat, broke its neck, and began to devour it raw. He was displeased with the taste, hypothesising that it might be more tolerable smoke-dried. (DW: The Two Doctors)

In the early 2000s, undercover Krillitane at Deffry Vale High School freeze-dried rats in order to eat them later. (DW: School Reunion)

In 2050, the Space-Time Manipulator released a couple of black rats in Gryffen Manor, frightening Darius Pike. While these type of rats were identical to normal rats, they were able to multiply once every ten seconds. (K9TV: Liberation)

Paradise Towers gained a population of rats after it fell into ruin. (DW: Paradise Towers)

On an unnamed planet, the Gyros destroyed rats which tried to eat the food the robots had harvested and stored. (TVC: The Gyros Injustice)

At some point in the 52nd century, rats would sneak into the Seventh Transept where they were eaten by the skulls of the headless monks. (DW: The Wedding of River Song)

In experiments
Rats were used in experiments as test subjects, since they could be bred quickly. In the late 19th century Magnus Greel created giant rats and set them loose in the London sewers to guard his underground laboratory. (DW: The Talons of Weng-Chiang)

Richard Harries used a pair of sibling rats in his experiments into shared brainwaves. The actions one rat learnt could be completed by the other through the process. The pair formed a collective memory. Cuthbert Simpson implanted them with nano-cams and set them free, using them to watch the residents. After Simpson was blinded, he relied upon the rats to allow him to see things and so kept them on his person. (EDA: The Banquo Legacy)

A lab rat was used by Owen Harper to demonstrate to the Torchwood 3 team the ultimately lethal effects of the "Sex Gas" on its host. (TW: Day One)