How does the title "Trifles" connect to the dramatic irony in the play?
Dramatic Irony: a state of affairs in which the audience and a few characters know more than most other characters.
Trifles: a thing of little value or importance
In the play Trifles, dramatic irony is achieved when Mrs. Hale, Mrs. Peters, and audience/reader begin to uncover the clues in the kitchen, the motives of Mrs. Wright in killing her husband, and the suppression of evidence by Mrs. Hale and Mrs. Peters. The men in the play remain oblivious, thus achieving dramatic irony throughout the play, with no real epiphany, realization, or downfall on their part.
Actually, the men have access to all the evidence in the kitchen, but since they think of women and women's domestic work as "trifles" or "trifling," they choose to ignore it. Instead, the men look upstairs in the bedroom and outside in the barn (two areas of male dominion) for clues. Obviously, they don't find any there.
So, the title is a form of verbal irony, both understatement and sarcasm, for the very important roles and work women play in the home. As Hale says, sarcastically, regarding the preserves:
Well, women are used to worrying over trifles.
If Hale, the Sheriff, and the Attorney would have understood women's work, the dead bird and the condition of the stitching, they might have found the clues and motive used to convict Mrs. Wright. But because the women realize Minnie's victimization by her husband and the sexist attitudes of their own husbands, they protect Minnie from going to a literal prison, having been living in a domestic prison for so long.