This is a great question for a number of reasons! Certainly this is a play when the women win against the men hands down! What is fascinating about this play is the way that the author uses dramatic irony so well to highlight the ignorance and chauvinism of the males in their judgement of the women.
The dramatic irony in this short play lies around the crucial fact that the men are completely unable to find a motive for the killing of John Wright whilst the women are, although they are disparaged by the men for concerning themselves with "trifles", which clearly in their opinion can hold no interest to their "serious" investigations.
You will want to look at how the men mock the women and infer that they know nothing, only concerning themselves with "womanly" activities. A key example of this, and one that is referred to again and again at various points in the play to highlight the irony, concerns the quilt that Mrs. Peters and Mrs. Hale find. Mrs. Hale says of this quilt:
It's log cabin pattern. Pretty, isn't it? I wonder if she was goin' to quilt it or just knot it?
Note then that the men descend the stairs, and the Sherrif repeats her words, drawing a laugh from the men. It is highly crucial then, that straight away after this, whilst Mrs. Hale and Mrs. Peters are taking up their time with "little things" as Mrs. Hale says, that they find the motive in the piece of crooked sewing, that gives evidence of "anger, or - sudden feeling", as Mrs. Peters reports Mr. Henderson saying. Note how Mrs. Hale describes what she sees:
Mrs. Peters, look at this one. Here, this is the one she was working on, and look at the sewing! All the rest of it has been so nice and even. And look at this! It's all over the place! Why, it looks as if she didn't know what she was about!
The women, by engaging in their "trifles", have found the motive that the men have been looking for, whilst they have been stomping ineffectually all around the house. The answer was under their noses all the time, but needed a woman's knowledge to piece it together.
Therefore the men throughout are disparaging and dismissive of the women and what they can offer to the "real" investigation that the men think they are engaged in.