If drainage is a big problem, you'll likely have problems with not only the pooling, but also simply growing good grass (the sod will die).

I think I would amend the soil and mix in some good organic material so as to make the soil natural better at drainage. Also, check the grade in those areas as well.

You can add topsoil and organic material so as to have a good 6-8 inches of rich soil, which should help.

If you add gravel, you will end up with very packed clay and gravel. If you add sand, you will end up with clay and sand cement. Build up the area by tilling in loads of organic matter and you will create a good soil that will drain of excess water, unless your topography is such that all of the water in the vicinity is running to that low point.

Clay soil can still drain well. If you have a few low pockets, then you will need to fix that before the problem can be solved. Even a slight change in elevation can create a pooling effect....like the previous follow up explains.

My local nursery said to add gypsite to clay soil, in addition to the other amendments.

I had the same issue.... spray the compacted clay with EarthRight. I used it in my problem clay area, and whola, no more problems. Neighbor put in a french drain, and he was fighting drainage all year. Last weekend he broke down and got the Earthright too.

Here is a link that might be useful: Earth Right soil conditioner