There were some HP ones linked to in another thread that were decent (like the one with Gilderoy Lockhart self-insert).

I think they can be fun to read in a way for reasons such as:
1. I kind of enjoy "fix the world" fictions and some of them add new or greater problems so it doesn't become an OPed thing
2. If the SI remains imperfect then they are stil identifiable and more so because its from our world and has our issues to bring with it and can comment on the things that don't make sense in the fictional world.

 

Incidentally, here's a self-insert done well, (at least for the opening section) from the LotR fandom:

Home with the Fairies By I-Mushi
We've all seen the horribly butchered and cliché girl-falls-into-Middle-Earth stories. So is it possible to make a semi-realistic one? Follow Maddie on her own adventure exploring Middle Earth as the opposite of a Tenth Walker.

It works for several reasons:

1. The inserted character is not aware that they're an insert. They remember the real world, but they know nothing of LotR.

2. The inserted character is mostly blank slate. She's kinda a generic modern girl. This helps you identify with her, because there's nothing in particular about her which jars against your sense of how the character should act. She's not hyperactive, jokey or whatever.

3. She doesn't join the Fellowship, or even really meet them. This prevents "fix it syndrome"

4. She doesn't speak the language, and it takes her a long time to learn it. This prevents things feeling too convenient.

5. The realities of living in a medieval world are confronted in detail. Similar result as #4.

6. She's not important in the global scale. The stakes are entirely personal. Again, avoids fix-it.

7. No love story. Avoids making the fic feel like wish fulfilment.

I haven't finished the fic yet so this might not all remain true. But it's true for a good part of it. So to summarise, a self-insert fic can work if:

- The inserted character isn't too savvy.

- The inserted character is bland enough to prevent them feeling like the author's representative.

- Fix-it fic is avoided.

- Convenience is avoided.

- Wish-fulfilment is avoided.

 

You do realize that many people read fiction to enjoy themselves, right?

This is one of those arguments that can go too far in either direction- yes, we want something plausible so that we don't trip over a moment of 'how could that possibly happen?', but there's usually a point for an adventure story to focus on a particular character. They ought to be the fulcrum of change, or at least the optimal point of view for seeing the scope of something. If Miss Nobody drops into Middle-Earth only to be discovered by a kind fishing family and taught the basics of language and a few skills to make her useful, yet she never finds her way into the main drama happening in the setting, we as the reader will be wondering 'who cares what this girl does?'

More directly, the question I'd ask would be 'why did the forces of narrative interference steal someone from another reality and drop them here if they aren't going to make a difference?'

 

So essentially the opposite of the protagonist in most fanfics.

 

Exactly. The problem with self-inserts is that they feel like outsiders usurping the position of protagonist which rightfully belongs to whoever is the canon protagonist. By giving the SI a contained story of their own, independent of the canon storyline but tangentially related to it, you can tell an SI story without stepping on canon's toes.

 

Unless the SI is not the POV and not the protagonist, but instead his appearance throws the canon storyline off the rails and now the true protagonist must deal with the changes.