There are three ways in which the literary device of allusion is used in Ray Bradbury's short story, "There Will Come Soft Rains."
First, the poem by Sara Teasdale is alluded to in the title and in the body of the story. Here is the poem:
There will come soft rains and the smell of the ground,
And swallows circling with their shimmering sound;
And wild plum trees in tremulous white,
Robins will wear their feathery fire
Whistling their whims on a low fence-wire;
And not one will know of the war, not one
Will care at last when it is done.
No one would mind, neither bird nor tree
If mankind perished utterly;
And Spring herself, when she woke at dawn,
Would scarcely know that we were gone.
Sara Teasdale wrote this poem in response to the destruction of World War I. It is a commentary that all of nature exists in harmony and purpose, and only mankind functions contrarily to this. Teasdale suggests that if mankind destroyed itself, nature would continue on, unaffected.
This is strikingly similar to what happens in Ray Bradbury's story. The house continues to cook meals, give reminders, recite poems, prepare for leisure time, and in all ways serve the masters who live there. The only problem is—the owners are gone. The only hint we have as to what has happened to them is the total destruction of the neighborhood, and the black silhouettes of a family on the siding of the house, their images presumably burned into the wall by a type of nuclear blast.
Another allusion in the story is the reference to the pagan god Baal. When the house is cleaning, all the dust and debris it sweeps up is sent into the basement, where it is put into an incinerator. Bradbury compares this contraption to the evil god which was introduced to the Jewish people through the evil queen Jezebel. In the Bible, the prophet Elijah discredits Baal by calling his worshippers into a type of contest in 1st Kings chapter 18. Here is the allusion in Bradbury's story:
There, down tubes which fed into the cellar, it was dropped into the sighing vent of an incinerator which sat like evil Baal in a dark corner. The dog ran upstairs, hysterically yelping to each door, at last realizing, as the house realized, that only silence was here.
Just like in 1st Kings, the worshippers of Baal were met only with his silence, in Bradbury's story, the house is met with the silence of its masters.
Another minor allusion is the reference to Picassos and Matisses that the fire consumed. This refers to Pablo Picasso's paintings, who was a Spanish artist famous for his cubist paintings, and Henri Matisse, a French painter in the post-impressionist era.