The Irony of a Dream
We see that Gelsomina comes alive in front of an audience early in the film. It's a dream come true to perform and be an artist; she has found freedom and self-expression. The irony is that she works with a man who does not respond to her outside of their performances. She is filled up when she is 'on stage,' but broken down by continuing on with a man who brutalizes her by never responding to her, nor attempting to meet her needs.
The Irony of Leaving
Gelsomina decides that she is leaving Zampano, and takes off for the city. She doesn't want to leave, but wants to be with him and for him to appreciate her. The irony is that by leaving, it is the first time we see him come for her. He could very easily have left her, but instead becauase she walked away he chooses to go out of his way to find her.
The Irony of Brutality
Zampano is the embodiement of brutality in this film. He represents the body and is a constant threat to hurt someone who gets tangled up with him, and he lords his strength over everyone he encounters. This is what makes him powerful. The irony is that because he leads with his strength he is quick to violence which leads to very serious consequences when The Fool is killed after being beaten by him.
The Irony of The Sea
Fellini uses the sea quite often in his films. At the opening of La Strada we find Gelsomina gathering wood during the day on the sea shore. Zampano has come to buy her, and she is overwhelmed to know that she will have the chance to see the world and be an artist. We see that the sea has given something to her soul. The irony is that the sea also takes away, and we see this contrast at the end of the film when Zampano is left alone on the shore in the darkness, drunk and alone after discovering that Gelsomina has died.
The Irony of Love
Throughout the film we see that Gelsomina has become quite fond of Zampano. At one point she even asks if he will marry her. We see a woman who's had no hope of love, find it, even in this brutal man. Zampano does not respond to her inquiry of marriage, and it is not until he discovers, years later that she is dead that we see that he has any feeling for the girl. The irony of love is that most don't know when they have it. It's only when it is gone that they recognize that it was right in front of them.