What allusions does Shakespeare use in Act III of Romeo and Juliet?

An allusion is a reference to a well known person, place, event, literary work, or work of art. Shakespeare often alludes to Greek and Roman mythology in his play Romeo and Juliet. He also alludes to stories or characters which would be well known to his audience. In fact, the idea for the play comes from a narrative poem by Arthur Brooke called "The Tragical History of Romeus and Juliet." It was familiar to his audience so it was not surprising that Shakespeare announces the eventual outcome of the tragedy in the opening Prologue. Thus, the allusions in Act III refer to common stories and mythology.

When Tybalt comes on the scene and Romeo backs down from the Capulet's challenge, Mercutio steps in to fight. He calls Tybalt a "rat catcher" and "Good King of Cats." This recurring allusion in the play refers to a character, also named Tybalt, in a well known medieval tale called "Reynard the Fox." The Tybalt from "Reynard" is also quick tempered and quarrelsome. 

After Mercutio is stabbed by Tybalt he launches into his famous rant condemning the two families for his demise and alluding to the plague, often known as the "black death." In his last words, he says, 

A plague o’ both your houses!
They have made worms’ meat of me.
I have it, and soundly, too. Your houses!

The Bubonic plague was a deadly disease which ravaged Europe in the 14th century and was still a threat at the time Romeo and Juliet was written.

In the beginning of Scene 2, as she is waiting for night so she may be reunited with Romeo, Juliet alludes to the Greek mythological story of Phoebus and Phaeton. Phoebus was another name for Apollo, the Greek god of the sun. The myth says that at dusk the sun would be hitched to a chariot pulled by Phoebus's horses and dragged across the sky. Juliet hopes that Phaeton, the son of Phoebus, will drive the horses to the west so that night will fall. Juliet says,

Gallop apace, you fiery-footed steeds,
Towards Phoebus’ lodging. Such a wagoner
As Phaëton would whip you to the west
And bring in cloudy night immediately.

A little later in Scene 2, the Nurse arrives with the news that Tybalt has been killed and Romeo banished, but initially confuses the issue with Juliet. Juliet makes an allusion to the devil and hell in her admonishment of the Nurse for not speaking clearly. She says,

What devil art thou that dost torment me thus?
This torture should be roared in dismal hell.

In Scene 5, Romeo and Juliet are in the girl's room. Juliet first makes reference to the nightingale, which she says is singing outside her window. It signifies that it is still night and Romeo should stay longer. Romeo contradicts her and says it is the lark, "the herald of the morn." The nightingale is linked to the tragic Roman myth of Philomela who had her tongue cut out and then transformed into a nightingale. In contrast, the lark has often been a symbol of joy, especially in Shakespeare's A Midsummer Night's Dream. Here, in an odd juxtaposition, the nightingale is good and the lark evil, since it represents the time for Romeo to begin his banishment.

Finally, in another allusion in the same exchange, Romeo refers to "Cynthia's brow" referring to the moon goddess (Cynthia is also linked to the Greek goddess Artemis). He is agreeing with Juliet that it is still night.