This might sound like a broken record, but there's no single style in Dubliners. What's so amazing about the stories is how many different styles Joyce employs to describe the various walks of life that represent a whole city.
He employs sentimentality when describing Eveline, journalistic precision when describing Mr Duffy, and street slang when rendering the conversation of Old Jack. If we're looking for one phrase to describe Joyce's style in Dubliners, one might be "personal naturalism."
That word—naturalism—is always associated with Joyce and with a lot of the fiction that was written around this time. It's a word that describes a way of writing that looks at the hard facts of modern life straight in the face and depicts them without exaggerated emotion or heavy interpretation.
Naturalism in literature was influenced by the rise of scientific research by such famous dudes as Darwin, as well as by recognition that modern life had been drastically altered by the industrial revolution with all its steam engines and factories and other big, gray things. The goal was to describe life as realistically as possible in the hopes that all of its dirtiness and misery would give rise to some sort of change for the better—more green and less grime.
Joyce's naturalism is a little bit like this, right? He doesn't shy away from showing us the worst aspects of Dublin life: the child molesters, the dark alleys, the drunk and violent men and their overbearing bosses. And he doesn't offer a lot of commentary about these things, but leaves us to come to our own (horrifying) conclusion. Contrast this with a writer who would say, "It was really Farrington's very mean boss, and not just his own violent temperament, that was to blame for the way his night went." Joyce doesn't do that: he just lays it out and asks us to connect the dots.
On the other hand, Joyce isn't just a naturalist. He's way too interested in the way that the individual mind works, and the way that emotions and all that other fuzzy stuff influence our decisions to just focus on the facts.
What would a story like "Araby" or "The Dead" be like if we didn't really get inside the heads of the characters and hear in great detail about feelings of self-hatred? What if the story just told us, "He didn't buy anything at the bazaar, and then walked home alone in the darkness"? It would still be a powerful image, but it's nothing compared to Joyce's revelation that the boy despised himself as a creature "driven and derided by vanity." Yowza.
That's Joycean naturalism: it's personal, it's powerful, it's super descriptive.