[…] the women mostly out of curiosity to see the inside of her house, which no one save an old man-servant – a combined gardener and cook – had seen in at least ten years. (1.1)
This funeral moment at the beginning of the story sets up the divisions that exist between Emily and the town. This sets the framework for Emily's isolation in life by talking about her funeral. Who's the most isolated person at the funeral? The corpse, of course.
After her father's death she went out very little; after her sweetheart went away, people hardly saw her at all. (2.1)
More proof of what we already know: Emily is isolated. Here we have two more major aspects of the isolation. Her father isolated her from men, and then the whole Homer Barron thing permanently isolated her form everybody (except Tobe), which seems to be what her father intended.
None of the young men were quite good enough for Miss Emily and such. (2.11)
In this line the narrator gives us both the vicious clucking of the town, and their assessment of her father's control of Emily. These factors work together toward Emily's withdrawal from the world.
So [Miss Emily] vanquished them, horse and foot, just as she had vanquished their fathers thirty years before about the smell. (2.1)
This moment gives us another big component of Emily's isolation. The smell was the beginning of the end. The interesting thing here is the word "vanquish." If Emily vanquished the lime-tossing guys, that means she conquered them. How, we ask, did Miss Emily conquer those men?
From that time on her front door remained closed, save for a period of six or seven years, when she was about forty, during which she gave lessons in china-painting. (4.5)
Was there hope for Emily here? Her former students are the ones keeping their children away from her. Perhaps those china-painting lessons were scary and creepy. Perhaps the waft of death reached them and allowed them to put together the bits of whispered gossip they heard around town.