Let's set the record straight
As much as it pains me to revisit this old hack, I have, nevertheless, as a nerdy English major, always wanted to break this down...

An old man, turned 98
He won the lottery, and died the next day--yes, this is indeed ironic
It's a black fly in your chardonnay--this is not in fact ironic. instead, it is simply a bummer.
It's a death row pardon, two minutes too late--yes ironic. also, a BIG bummer.
And isn't it ironic?
Don't you think?

It's like rain, on your wedding day--not at all ironic. disappointing and a pain in the ass, yes.
It's a free ride, when you've already paid--check. ironic.
It's the good advice, that you just didn't take--this is too vague to really be anything. hubristic? maybe expand on the specific situation... ie, "don't order the chardonnay here, there are always flies in it." and then you order the chardonnay and there's a fly in it?....
And who would've thought? It figures

Mr. Play-It-Safe, was afraid to fly
He packed his suitcase, and kissed his kids goodbye
He waited his whole damn life, to take that flight
And as the plane crashed down he thought, "Well isn't this nice?"--ironic. but more aptly: doomed.
And isn't it ironic?
Don't you think?

It's like rain, on your wedding day
It's a free ride, when you've already paid
It's the good advice, that you just didn't take
And who would've thought? It figures

Well life has a funny way of sneaking up on you
When you think everything's okay and everything's going right
And life has a funny way of helping you out
When you think everything's gone wrong and everything blows up in your face--this is more like karmic retribution, than irony. not sure what it is doing here.

A traffic jam, when you're already late--this is not ironic. it doesn't matter if you're late when you're late! it's a great time to be late. or to have a traffic jam! kick back, relax!
A "No smoking" sign, on your cigarette break--again, needs more specifics. just go someplace else.
It's like ten thousand spoons, when all you need is a knife--ten THOUSAND spoons?, that's a fucking freak show.
It's meeting the man of my dreams, and then meeting his beautiful wife--ennhhh, more like, wistful? or, DOH!
And isn't it ironic?
Don't you think?
A little too ironic?
Yeah I really do think

It's like rain, on your wedding day
It's a free ride, when you've already paid
It's the good advice, that you just didn't take
And who would've thought?
It figures

Life has a funny way of sneaking up on you
And life has a funny, funny way, of helping you out
Helping you out

PHEW! GOD, that was bugging me for the past 13 years!