It is helpful to remember that irony is, first and foremost, a literary device. The Greeks coined the term Eironeia (literally "dissembling") to denote a rhetorical device that orators and dramatists could use to make a point. It was no more intended to represent observable, real-world phenomena than similar terms like synecdoche or metonymy or anaphora.
That's why it's so difficult to find actual, real-life examples of situational irony (as opposed to simple sarcasm) in action. It's supposed to be a poetic device, not a practical observation. It is the rarity of true cases or irony in the real world that causes them to have such resonance.
By the original intention of the term, an ironic situation is a reversal of expectations that has been deliberately crafted to make a specific moral or political point. Unfortunately, too many people simply stop at the "reversal of expectations" step, so that any situation that is unpleasant or frustrating is translated as irony.
The situations Morisette describes in Ironic are almost exclusively prosaic and inconsequential. "Rain on your wedding day" may be annoying, it may be disappointing, but there is no lesson to be learned there. "A free ride when you've already paid" may be bad timing, but it's not evidence of a moral failing. There is simply no point to the story.
If you're looking for a single word to encapsulate all of the examples Morisette uses, the best I can think of would be "unfortunate".