As the other answer to this question suggests, Joyce does not use stream of consciousness narration in Dubliners; he uses that technique most famously in his later novel Ulysses, which depicts the inner monologues of Leopold and Molly Bloom and Stephen Dedalus as a tumbling rush of associative language, a sequence of thought that moves along like a stream (especially in Molly's chapter, which dispenses with the convention of complete sentences and delivers thoughts in a seemingly endless stream without punctuation).
However, Dubliners is well-known as an example of another important narrative technique: free indirect discourse. Using this narrative technique, Joyce's third-person narration adopts the style of his character's consciousness, resulting in a text that is not directly "inside" a character's head but that nonetheless mirrors his or her distinct way of thinking or speaking. A famous example of Joyce's free indirect discourse occurs in the very first sentence of "The Dead": "Lily, the caretaker's daughter, was literally run off her feet." Now, as the subsequent narrative reveals, Lily wasn't actually knocked down, as the first sentence suggests. Rather, she's merely so busy with receiving guests to a party that she feels as if she's been "run off her feet." The sentence becomes an example of free indirect discourse because the third-person narrator adopts Lily's distinctive use of language to convey an idea, thus giving us a snapshot of her consciousness.