Constructive criticism really tends to break down into three important classes in my mind. One is the setting (virtual or actual) of the criticism, another is the expected amount / types of discussion, and the last is the audience in question. Sometimes, it seems like people are blind to how these classes work.
Let me explain. If you're in a creative writing class at most schools, you're probably told that constructive criticism has to have at least one positive thing before you get into the negative. Why do you do this? Well, first of all, it's a balm on the author's ego to get something positive about their work, and especially in a group setting, it encourages people to try to find something good about a piece. It also encourages the author to actually listen to the advice when it's not so positive. So, creative writing tends to get at least some positive encouragement.
However, an art teacher in a university, a coach, or a music teacher is likely to be less kind when talking to a student. One friend of mine had a roommate who was told she shouldn't be at a certain university because she was no better than the average ameteur choir. In the setting of that school though, this was considered a way to drive the students into trying harder to excell. Arts and sports often can get more negative and harsh criticism.
Settings: Of course, all these 'classes' are intertwined. Trying to examine them separately will immediately bring up the other classes, so it's not easy to talk about an example with only one thing that applies. For example, your setting will of course determine what audience you expect and what your audience expects. It'll also determine what the sort of expected criticism levels will be and who the criticism will be intended to help. A newspaper movie review isn't intended to help a director shoot a better movie, and won't help a film student analyse it. It will, perhaps, help a reader decide if they want to watch the movie.
Semi-recently, the setting issue came to my attention. Someone had been giving strange comments, and when questioned about it, they burst into a rant about how her porn was the worst writing they had ever seen. The phrasing of the rant looked like something you'd see on a Mary Sue smiting journal - extremely abrasive, rude (declaring that it was bad fanfiction by a fluffy twit,) and written with an air of authority. There was two problems though, the person was incorrect (in my opinion) on some of their conclusions, and they had no interest in discussing their issues. As a setting, an author's journal can be a place to give criticism, of course, especially if the author requests it. However, blunt statements of 'fact' with no interest in discussion isn't exactly a criticism style that encourages the reader to listen. Being wrong when you're trying to write from a position of authority and not being willing to discuss it is also likely to make people ignore you. The person had misread the setting as to how much criticism was expected (thus the odd passive aggressive comments,) had made incorrect conclusions, and then attempted to evade discussion via comments like "it's just my criticism." Needless to say, the comments (irregardless of the truth of part of the claims,) didn't go over well.
Expectations: Now, an art student may want to get more gentle criticism, but they are unlikely to get it. As a creative writing student, I often wished I could get (and give) more harsh criticism. The expectations of the situation, however, dictated what sort of criticism was given. Generally, a teacher has more leeway in defining what the degree of criticism in their classes. This makes perfect sense since the teacher should be someone with authority in the field in question. This also puts a burden on the teacher to encourage the type of constructive criticism that they desire.
On the other hand, if you don't consider a person to be an authority figure, attempting to define the expectations of criticism can be highly insulting. (By defining, I mean, for example, ranting that a reviewer on their own ground must say something positive about everything.) This plays back into the setting question and the audience question. If you're expecting criticism on a story from the point of view of a creative writing student, and instead get a harsh editor's response, you're going to be quite startled. If the setting, though, is one that such editing is normal, you're the one who's out of sync. If you are writing for friends and writing a quick drabble, your audience expectations don't fall in line with a harsh editing for grammar. A quick doodle of an in-joke isn't normally something that you expect to get a detailed rant about proper sketching of hands.
One reason why I hate most recommendation lists is that they tend to be vague. The sixth or seventh "Story X is great. I can't spoil the ending, but I loved it. The characters are well written," doesn't give much for the author or the reader. I can't tell if the recommendation list author likes what I like, and I can't tell if their "well written" is my "flat ciphers flailing at cliches." I need more information to make a decision about reading a story than just someone saying they liked it. However, the expectations of most lists is usually that they'd provide a link, some idea of the content, and sometimes a warning for iffy content. As a reader, I want to see the warnings for the story (both to look for personal squicks and kinks,) and ideally some idea of what the writing style is like (no caps, florid writing, trick endings would make me cranky.) As a writer, I'd like to see comments about the plot, characters, and writing, ideally with support instead of a vague "I didn't get Bob." My expectations, however, do not fall in line with the expectations of most lists.
Audience: I like to read romance novel reviews. Particularly of awful books. The good humored discussion of poor writing, bad sex scenes, etc. can be quite fun. I don't actually read romances, but if I did, I've got a vague idea of books that I'd probably hate. Often, you'll get people wailing that these reviews are cruel to the authors. This is an audience issue. If you go to a romance novel reviewing site, and they say they are writing reviews for readers / for their tastes, the audience is not the author. Of course, the author may learn, for example, that a profusion of roosters in the text can get silly instead of erotic. But, they are not the intended audience.
Constructive criticism of a text for a reader ideally should support the claims (bad writing because of X, good plot because of Y.) It should also highlight why a reader may or may not want to read a story. It's constructive for the reader because it tells you why the author did or didn't like something. If you agree, you know to look for highly rated stories. If you disagree, the low ratings may be exactly what you want. A good reviewer will state how personal issues may change the criticism, and you, the reader, can take that into account. Often, you don't need to state your authority when doing this sort of criticism. The reader can decide how much to believe or disbelieve your conclusions, and follow your advice accordingly. The famous thing about a movie reviewer not needing to make a movie applies here.
Constructive criticism for an author, on the other hand, is more complex. The author may have specific desires for what kind of criticism they want. If they want, say, plot criticism, telling them about every grammatical mistake will be irritating, even if it is helpful. The setting matters. You don't want to give a snarky reader orientated criticism in the author's space, since really, it's too late for the reader to be warned. If you have serious criticism, you need to make sure that you're correct and be willing to have the author get irritated if they don't want that sort of criticism.
Now, when reviewing for an author, you don't need to be an author, but, due to the fact that some authors can be defensive, it's good to show that you know what you're talking about. Complaining, say, about a character being alive is going to discredit your authority if it turns out that the character has every reason to be alive. One person was upset that I had a character who was "scary" and abrasive and took the trouble to find every time that she had been assertive in the story. The problem was that she was supposed to be scary and abrasive. By complaining about it, the reviewer just revealed (in my opinion) sexism, and a lack of understanding of the character. Another reviewer complained that the dialogue (of another story) was very casually written with lots of contractions. They thought that it sounded realistic, but that it was poor English. This undermined the authority of their criticism.
The audience intended for the work (and where it's placed,) will also change the criticism. This is simple logic. Obviously a fast scribble isn't fine literature. Obviously a quick doodle isn't high art. However, when the doodle is propped up in a gallery, then it's more acceptable to start thinking about the ways it is (or isn't) high art.
So audience matters as to who the criticism is aimed at and who the work is aimed at. It also matters as to who will be seeing the criticism. If you don't take all of these into account, complaining about the criticism (or the work) may go over poorly.
So, comments? Ideas? I haven't seen a lot of places really breaking criticism down into these categories, so that's why I typed it out.