How does the story "A Rose for Emily" exemplify modernism?

The story exemplifies the tension between the New South and the Old, between a world of certainty and order and the much more chaotic modernist world, where all the old uncertainties have been undermined.
We see this tension illustrated right throughout the story. Miss Emily represents the old world, and she is placed on a pedestal by the townsfolk as the last surviving link to a supposedly more gracious past. Yet modernity cannot be stalled forever. The town needs to function and taxes raised for that precise purpose. This means that Miss Emily must contribute her fair share. However, because of her exalted status in town, Emily is able to cut a sweetheart deal with the local authorities which means she doesn't have to pay any taxes. That's definitely a win for the Old South.
Another win is Miss Emily's being able to buy rat poison without specifying the precise purpose for doing so. Modern laws require her to do precisely that, but Emily triumphs in her brief battle of wills with the pharmacist and once again gets a pass.
Modernism eventually triumphs over the forces of the Old South, but by then it's too late. It's only when the source of that revolting stench emanating from the Grierson residence is finally revealed that the modern world triumphs over the old. We can see this as a prime illustration of a key aim of modernist literature, namely to reveal what's really lurking beneath the surface of the ostensibly ordered, respectable world in which we live.