Busts tend to be my favorite system
in general, although there are situations where other methods might be smarter choices - such as a very fast-paced RPG that has a lot of characters (and NPCs) but doesn't put a large emphasis on dialogue; for a game like that, simple bubble dialogue that doesn't stop the action could be a really good design choice.
Some games wisely combine two or more different dialogue systems, too. For example, I believe
Chantelise used non-intrusive Text Bubbles when interacting with characters in town (as well as for hints, etc., while battling in dungeons), but used full Bust + Text Box dialogue during story scenes.
The single best dialogue system I've ever seen in a game is
Oxenfree, which kind of bucks all of the normal dialogue techniques and goes for something far more natural:
Here are my thoughts on each system you mentioned:
1) Text box + those square faces on the left side. (standard-like)
I'm not crazy about this in general, but it's a good start for designers who are very tight on funds and/or have a large ensemble cast of characters, since faces tend to be a lot cheaper/quicker to make than full busts (and can be done in RPG Maker using the Face Generator). It also has the advantage of taking up less screen space than the busts, which is nice if you have a lot of onscreen choreography happening behind the dialogue.
It's best when multiple faces can be shown for different emotions/tones (although most of RPG Maker's generator mouths are terribly ugly).
2) Busts/Portraits + text box. (well-known. No need for introduction)
Generally my favorite; busts tend to not only be expressive, but really give a sense of who the character is (based on their clothing and posture). I like busts a lot. The downsides (besides the time/money to make them) include the large amount of screen real estate they take up, and the amount of time they take for the player to visually take in.
This means they're best for scenes where the dialogue is the main thing happening (rather than complex choreography or player-driven action). But even for regular text in the middle of, say, town/dungeon exploration, they are not that bad.
Bubble dialogue definitely has its niches. Where it doesn't require player input to move the dialogue forward, it becomes a much more effective system than any kind of text box, making it the perfect text technique for games where you can talk to characters in the middle of dungeons, or where there are a lot of NPCs around town without too much important to say.
Additionally, because bubble dialogue takes up very little screen space and can easily and identifiably attach itself to a character onscreen, bubble dialogue is the way to go if you often have multiple characters saying different things at different times, or regularly interrupting each other - or even if you just have a lot of choreography onscreen that's happening during story scenes and you don't want to pause the action for several seconds for the player to read three text boxes before continuing.
The downside of bubbles is that (unless you also include the character's face in the bubble, which reduces some of the advantages) you are relying on sprites to convey the character's actual look, tone, and expression - and sprites tend to be poor at conveying all of these. If you can design detailed and expressive sprites, that move in realistic ways and show off emotion with either facial expressions or body language, then you can get around this and create something truly brilliant. That's hard to do though - consider
To the Moon, which had a truly beautiful story but was let down just a bit by its reliance on sprites and text bubbles.
4) Text box + sprites (like in this video)
I don't like this method at all. It's interrupting gameplay (and perhaps choreography) in order to display text boxes alongside dinky little sprites. Even if the sprites are well-made and expressive, it's nearly impossible for them to match the kind of tonality and expression that either faces or busts can provide. I really don't see a single advantage to this technique, aside from not bothering to create any form of face or bust - might be good in extremely low-budget indie games made outside RPG Maker (where Faces are already available) and very little emphasis on dialogue.