It is not really a swear word, but I would tell my kids not to say that. It is not a “nice” word, although not extremely rude either. Let’s say it is below average, polite conversation, but not very offensive either. You would most probably not write it. You can use it between friends, or in family (not counting the kids). The nuance is more one of disgust than anything else.
For example, if someone did something really bad to somebody else at work, you could react with c’est vraiment dégueulasse, c’est vraiment dégueu (also commonly used) to show your complete disgust and your contempt for the offender. Disguting with the proper tone in English is quite close, although the French dégueulasse is less advisable (disgusting and dégoûtant (see below) share a common root (but I did not verify)).
An even less advisable word would be gerbax (“puke inducing“?), and this is definitely reaching into slang. Going in the other direction, an alternative could be: dégoûtant, and here we have crossed into more polite territory, polite enough for any conversation. Répugnant works too. As do révoltant and écoeurant. These last three are quite-middle-of-the-road, with slightly different nuances each time. Sale and horrible would be possible, but too weak and tame. Immonde is possible. So is infâme, which is now quite formal.
I would say that “dégueulasse” is not a swear word, but it is offensive, although it is commonly used in casual situations. It litterally comes from “dégueuler” which is the offensive word for “puke”.
French has this concept of “gros mot” which covers vocabulary of a certain level that typically includes swearing, vulgarities, and rude or offensive words. “Dégueulasse” is not a swearing word but it is vulgar, so I would definitely label it as “gros mot”. Like Frank said, kids would probably be reprimanded for using it.
Many words that end with the suffix “-asse” are pejorative at various levels: rêvasse (familiar), vinasse (slang for bad wine), tignasse, pétasse, pouffiasse (down right insulting), etc.
I wouldn’t say this is a particularly offensive word, rather I consider it familiar language. It would certainly sound strange in an official speech, but I have heard this word in daily use and never noticed people being frowned upon for using it. Especially, pas dégueulasse is being used, meaning not bad at all.
Sure, the word does bear a negative connotation and can be used as offensive, especially if reinforced by constructs like gros dégueulasse or espèce de dégueu. But this is true of many words which are nevertheless appropriate in unofficial speech.