Okay, here are the things in the lyrics to that song that are supposed to be ironic. You tell me which ones are ironic by any definition. Actually, I'll tell you why they aren't.

Quote:
An old man turned ninety-eight
He won the lottery and died the next day

The expected result of winning the lottery is ... nothing, really. The expected result of being 98 is death before long.

Not ironic.

Quote:
It's a black fly in your chardonnay

The expected result of having chardonnay is, again, nothing.

Not ironic.

Quote:
It's a death row pardon two minutes too late

The expected result of being on death row is being put to death.

Not ironic.

Quote:
It's like rain on your wedding day

The expected result of having a wedding is getting married.

Not ironic.

Quote:
It's a free ride when you've already paid

The expectation of paying for a ride is getting a ride. This actually could be ironic if the story was further established, like maybe if there was a struggle to find funds to pay. But it's not.

Not ironic.

Quote:
It's the good advice that you just didn't take

I don't even know how to approach this one. I suppose there's not any direct result from receiving advice.

Not ironic.

Quote:
Mr. Play It Safe was afraid to fly
He packed his suitcase and kissed his kids good-bye
He waited his whole damn life to take that flight
And as the plane crashed down he thought
"Well, isn't this nice."

This one's real close. Let's give her the benefit of the doubt and say:

Ironic.

Quote:
A traffic jam when you're already late

Honestly, I'd say that the expected result of starting out to work late is hitting a traffic jam.

Not ironic.

Quote:
A no-smoking sign on your cigarette break

Okay, that's just being stupid. One would assume that if you're taking a cigarette break, it because you're not allowed to smoke where you were to begin with. Why would you go somewhere that also doesn't allow smoking.

Not ironic.

Quote:
It's like 10,000 spoons when all you need is a knife

The expected outcome of needing a knife is searching for a knife. This approaches irony, but:

Not ironic.

Quote:
It's meeting the man of my dreams
and then meeting his beautiful wife

Again, meeting the man of ones dreams has no explicit expected outcome.

Not ironic.

These things are all unfortunate coincidences, but are mostly not ironic.

Feel free to say I'm wrong, though.