Rain on your wedding day isn't, in itself, ironic - BUT a song titled "Ironic" getting the very concept of irony wrong, that is the very core of irony, so... well done.
The whole song is ironic, if you imagine the singer is supposing a story tale quality to her life. It's unexpected (and therefore ironic) if I expect my life to turn out perfectly, but then other normal and expected things occur that prevent my fantasy from materializing.
I knew exactly what the song would be before I tapped the link... But I definitely wasn't expecting the video with it, that was incredible. Managed to improve on what was already a masterpiece.
As others have pointed out - there are plenty of things in the song that actually are ironic. So what's ironic is people keep complaining about the lack of irony in a song called "Ironic" when it's actually got irony in it.
I forget who it's from, but this attitude reminds me of the quote: "Most of the ingredients in this soup aren't even turds, so what are people complaining about?"