Nobody else has started posting spoilers and thoughts here, so I thought I'd be the first :)
Firstly, it appears that I can now see slash in everything. Sorry, but you've been warned.
Secondly, perhaps I should make it clear that I really liked this book. I thought it was far better than OOTP, and I loved a lot of things about it. People seem to think that if I ask questions/comment/nitpick it means that I don't like something. Actually, it means quite the opposite. If I didn't like something, I'm not interested enough to do this. The more I do this, the more I liked it.
1. What happened to the Marauders in this book? They took a fairly central position in OOTP (if I'm honest, one of the things I liked about the book, which is still my least favourite) and yet were dropped from this one like stone.
Sirius - dead and practically ignored. No one mentions him much, except to suggest Sirius/Tonks. We still don't know what the bloody hell the veil was, the mirrors have been ignored and Harry's grief is pretty strongly submerged.
James - at the end of OOTP, Harry's sitting around feeling like his father was pretty nasty, and feeling miserable about it. This book - no James at all. Huh?
Lupin - I mean, come on: he's still alive. Couldn't we just have a go at putting him in the book a bit more? Apart from the fact that it clearly takes him so long to pair up with Tonks because he's got to grieve Sirius first.
Peter - is put in near the beginning and then totally ignored, just as he was pushed aside in OOTP. Well, come on, tell us - what's his motivation? And what's with the slashy subtext about him and Snape?
2. Dumbledore's hand. Although he apparently did it fighting a horcrux so we know the general gist of how it happened, he several times promises to tell Harry the whole tale and never does. Why not? Because he'll have to admit that it's slowly killing him?
3. It's too easy for Snape to be evil now. Don't think there's any way that he can be, really. Am STILL burning to know why Dumbledore trusts him (and am not going to suggest, no no no, not going to suggest a slashy reason for it). But, you know, there are several ways (hand, potion, old age) in which Dumbledore seemed to be dying before Snape killed him, and various suggestions that Dumbledore wanted it done (conversation in forest, the pleading which couldn't be to prevent "the next great adventure" given D's earlier comments, his insistance that Harry find Snape). And Snape yet again protects Harry from the Death Eaters and doesn't kill him when he easily could.
4. I wish Dumbledore hadn't spent the whole book saying "I'm really clever, me" - or variations on a theme. It began to annoy me.
5. I felt there were too many new characters introduced and too many old ones dumped or used very sparingly. Yes, okay, we've got Kingsley Shacklebolt moved to the PM's office - but then not mentioned again. Lee Jordan has now totally disappeared. Even Neville and Luna are given bit-part roles, despite being major players in the last book (and okay, I never liked Luna much, but still...). Wasn't there someone who had at least been mentioned before to make Minister? Or to bring in as Potions teacher? Why not Lee J as the Weasley Twins' assistant? What was the point of McLaggen?
6. Lily's eyes. Have some hypnotic quality about them, presumably. Or rather, Harry's inherited eyes like Lily's with the same sort of power. Which helps him get Slughorn's memory out of him, presumably, given that there's a big bit about how Slughorn can't stop looking into his eyes even when he wants to turn away.
7. Felix Felicis. A useful device, just as the timeturner was. Similarly hole-filled and difficult to use as an idea. It worked, but she might have been better off without it. (And, incidentally, I couldn't help but notice the destruction of all the other time-turners. Let's get rid of an awkward plot device that has served its purpose but is begging for too many other uses.)
8. And of course I'm disappointed by the threatened lack of school story side to the last book. As a s.s. fanatic, what are my feelings supposed to be now? I feel robbed.
9. It is entertaining that the title translates to "Harry Potter and Severus Snape", though, isn't it?
10. Doesn't Harry look at his Potions book and find that it's 50 years old? This put me off the scent of Snape as HBP, after he'd been my first thought, so surely I haven't dreamt this. But if the book were that old, then Snape would have had to have it second hand himself, because he certainly wasn't at school 50 years ago. That would make him just a few years younger than Voldey. Which he isn't. He's significantly younger and the same age as the Marauders.
11. Tonks. I'm uncomfortable with the characterisation of her in HBP. There are a couple of problems I find in her inability to do magic whilst upset. Firstly... in PS/SS, at the beginning, wasn't it suggested that Harry's magic came out precisely when he was upset or angry? (Even magic of an unnervingly metamorphmagus kind: he was upset that he was going to have to go to school bald, and his hair grew back.) Okay, maybe it becomes less controlled - you could have had fun with Tonks ending up looking like all sorts of things as a result of not being totally in control of the metamorphasising.
But I'm also unhappy that we seemed to get a cool adult female in OOTP, and she turns into a bit of a state because she's in love and it's all unhappy. Meh. Uncomfortably close to a bit of a stereotype. Why not have Lupin having those problems? Why pick on the female to get all upset?
12. The Half-Blood Prince business. Now, this is slightly more arguable, but if James and Sirius used Snape's spells against him, aren't they likely to have been reading his books? And therefore, aren't they likely to have come across the fact that he's calling himself the 'Half-Blood Prince'? And come on - if they knew that, then Lupin would have known too, surely? And mentioned it to Harry when he asked?