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Note: This isn't an attack on the ships themselves, nor the shippers. It is simply my thoughts on the ship. That said - feel free to tell me if you disagree (or feel I was harsh).
In this essay, I'm going to focus on the two most popular Draco ships: Hermione/Draco and Harry/Draco. (Though I'm not sure if Hermione/Draco's as popular as it used to be).
The main problem with those two, in my opinion, is that they have both been hurt by Draco immensley.
According to Draco's current philosophy, Hermione is a no-good "Mudblood," with no-good parents. She shouldn't be allowed a wand - and perhaps not even allowed to exist.
Lily Potter, Harry's mother, was also supposedly a no-good "Mudblood", whose death was well deserved. She and James asked for it, and Draco has not hesitated to remind Harry of this now and then.
For Hermione/Draco, the counter-argument to that I often see is, "but he'll be redeemed!" With Harry/Draco, it's often "they work past that."
There are people who will work past that, yes. But again, two issues arise.
First, Hermione takes pride in who she is, and Harry's parents are the meaning of his life. They died to save him. So I don't think either of them are going to get over that, especially Harry. His parents are his world.And Hermione, while she doesn't seem particularly close with her parents, does seem to think highly of them. People who have stable relationships with their parents aren't going to date men who would gladly have witnessed their murders - even if they've "come clean." That's not something you'll forget.
Second, Hermione and Harry are two of the most unforgiving people we see in canon, except for Sirius and Snape. Harry pretty much stalked, and even nearly murdered (albeit on accident) Draco throughout HBP. Hermione, to her defense, didn't take part in the stalking, but she led Umbridge into the forest (which, while that was self defense, was still disturbing because of the lack of fear later. Nearly killing someone isn't something to shrug off, no matter what).
Even if Draco did redeem himself - and to be honest, after HBP I'm not sure he won't - he isn't going to ever be a nice person, and I definitely don't ever see him being forgiven enough by Harry and Hermione for a stable romance.