I have a whole document of headcanons I keep on my Tumblr, but I figured I'd put them here. It's basically what I keep in mind while writing all my Undertale stuff. ( I wrote this before I started my Gaster story, so a lot of the headcanons here are gonna be plot points, but eh.) 

SANS AND PAPYRUS STUFF

Sans doesn't have a full formal education; he quit school when his parents dissolved so he could take care of Papyrus, which means he has roughly up to a 9th grade education. He was able to self-study with books from the library and the dump and look at what Papyrus was studying, but he would never call himself academic.

On the surface, though, he does finally end up going to college to get his doctorate. In Physics, obviously.

Papyrus did finish school, with excellent marks. Sans is innately better with academia, which is why he was able to work with Gaster, but Papyrus is a much more diligent student.

Asgore insisted Sans take the role of Judge after he fought Green Soul and came to him to say that they didn't deserve to be hurt–they had both been startled by each other, and Sans was quick with his attack. 

 He had to do train with the Royal Guard before he could officially be Judge. He hated every minute of it, and it took him a year longer than it should have because he wasn't motivated. He barely got out before Undyne was assigned head of the Royal Guard, which meant a) she doesn't know what his job is, and b) she would have probably murdered him for being so lazy. He doesn't mention this to Papyrus; he doesn't want to encourage his baby brother to have any part in fighting.

Papyrus is AMAZING with languages. Which makes sense, since it's the same method of thinking as solving a puzzle. He learned a bit of Gaster's language during the little while he knew the scientist, and he picked it up far faster than Sans did. When he gets to the surface, the Romantic languages are his favorite because they all have the same structure. ("It's like a LANGUAGE PUZZLE!!!")

Sans can sew pretty decently; he made most of Papyrus' battle body, and he has to adjust most of the clothes he buys since he's so short. Their mom was one of the top seamstresses in the Underground (and main breadwinner in the household, but we'll get to that later) before she dissolved, and a good deal of her bonding with him was teaching him to sew as well, starting with hemming. He had to sell her machine after their dad dissolved, but he still manages all right with just a needle and thread. 

GASTER STUFF

He couldn't communicate when he first moved underground, just vague gestures. Toriel was the one who taught him the language; she had a book on Skeletons, and it vaguely mentioned the Gasters' language. As a result, his speaking style is fairly close to hers, if more disjointed and animated.

Unlike Alphys, Gaster isn't technically a doctor. The skeleton village really didn't have a university, and the Underground couldn't put one together until many, many years after after establishing New Home. But Asgore insisted on him becoming Royal Scientist after he mentioned his idea for the Core. Since it was desperate times, he agreed, but it wasn't until about 700 years later that he felt comfortable introducing himself as "Dr. W.D. Gaster." Even so, he vastly prefers to simply be called Gaster.(His name sounds silly when spoken aloud, even if it's lovely when it's signed.)

He enjoys Asgore's company and is eternally grateful for his job, but Toriel is the royal he was closest to. They would meet up once a week and discuss the books they had read over tea and pie. At one point, he translated a bunch of skeleton stories and made a book for her, which she loved.

(Later, Toriel would find the book collecting dust in the back of her shelf as they cleared out to move to the surface. She insists Sans takes it, saying she has no idea where she got it from, but, as a skeleton, he should find it interesting. He knows exactly who it's from, but says nothing and instead reads it to Frisk, Papyrus, and Undyne during a sleepover.)

MISC. STUFF

Asgore has a really bad habit of assigning roles to people out of desperation: Gaster, Alphys, Undyne, and Sans are all good indicators of this. The good news is that he has a good eye for character. Some take well to it, such as Undyne, some are wary but do it, like Sans and Gaster, and others completely can't handle the pressure, like poor Alphys.

Speaking of Alphys, if Gaster hadn't been erased and she had continued being his assistant before he handed off the mantle, she would handle the stress of being Royal Scientist and the complications that came with it much better. 

Before Gaster's erased, she shows Mettaton to Asgore, who then calls in Gaster. Gaster offers her a position as his assistant, and, knowing full well that he's quite old and should just start thinking of the future, trains her to become his successor (without putting pressure on her; he can tell she has a nervous disposition.) He also takes care of a majority of the DT experiment's letters. After Gaster's erased, the timeline changes: after showing Mettaton to Asgore, he shoves the title on her, and she's understandably stressed out.

Skeletons are naturally reclusive, much more than monsters; the village was clan-based, so families tended to stay to themselves. When Sans was young, his parents stayed to themselves, and it wasn't until he went to school that he started to really be socialized with children rather than adults. Papyrus had less of an issue because Sans was there, but their upbringing still left them a little socially awkward at school. Sans lost a lot of his awkwardness when he started working; Papyrus...is still Papyrus.

Gaster was never one for going out or the grand events that monsters tended to have, but he was still rather sociable during the first few centuries of being Royal Scientist. It was against his nature, but he managed. After Toriel left, though, he went back to his reclusive nature and only talked to Asgore for years after that.

Both Toriel and Sans have the most endearingly terrible laughs. They manage to hide them well with demure giggles and "heh"s, respectively, but when something really funny catches them off-guard (or when they go on a pun rampage), it's the most god-awful cacophony you've ever heard.

ADDENDUM:

Figured it was worth mentioning, but in my head Sans looks like Insomnia!Sans (small and wiry rather than all the floof) and sounds like Cr1tikal (especially his laugh.)

Papyrus looks like Papyrus and sounds like Skeletor (paper-mario-wiki on Tumblr is my fave Papyrus, and also does an excellent Sans which is basically a modified Cr1tikal).

Gaster is super spindly and more than a little Jack-Skellington-like. I don't have any really good visuals to suggest. Or voices to suggest. 

 ¯\_(ツ)_/¯

THAT ALL SAID, imagine whatever you want! If you want Bara!Sans, then by god you imagine that Bara!Sans.