There was a young woman who lived in a hat.
She could bark like a dog and meow like a cat.  

The Young Woman Who Lives in a Hat is the title character of a lesser-known Nursery Rhyme. As her name suggests, she lives in a giant hat (a brown top hat to be precise). She does continual and perplexing impressions of dogs and cats. She kindly points Kermit to house of her neighbor, the Old Woman Who Lives in a Shoe, in a Sesame Street Newsflash. Her house is located between the Young Man who Lives in a Glove and the Old Woman Who Lives in a Shoe.

Incidentally, the same puppet was used in another News Flash, this one involving Jack and Jill. Here, the character's name is Francine, and she was performed once again by Richard, using virtually the same exact voice. She volunteered to go up the hill with Jack, but since her name didn't rhyme with the word 'hill', she was politely rejected.