Likely damping off. Spray with a weak solution of fungicide.

Sounds like you have been keeping them too wet. White spider-web type fungus will grow on the soil surface and you can also get a greenish mold. Both are signs of the mix kept too wet. Let it dry out more and it will disappear.

There is a FAQ about it over on the Growing from Seed forum if you want to read it.

Dave

A weak fungicide or a 10% H202 solution should get rid of it and letting them dry out between watering should prevent it from coming back.

I mist the soil with chamomile tea (it has antifungal properties) to prevent fungus and damping off. I don't know how effective it would be once the fungus is already established though. A box fan to increase airflow can help too.

Here is a link that might be useful: Seed starting (blog)

Another mild remedy is a weak solution of potassium permanganate - really weak, barely pink, which may be not so easy to achieve, you may need to make serial dilutions. After which I carefully "mulch" this moldy soil around the plants with a little more fresh potting soil. Works for me.

I have that growing on top of the soil on many of my plants as well. Seems harmless and none of the plants have had any trouble with it. Of course, that doesn't mean that the fungus is good - just doesn't seem to be causing any problem with the plants.

Here is a link that might be useful: My Garden Blog

The best solution is to avoid getting it in the first place. Most well over-water their seedlings even when they think they aren't so cut back on the water. And improving the air circulation around the plants is a great help. It is the primary reason why many use a small fan set on low just to keep the air circulating and cooler around the plants.

For fungus to grow it needs wet, warm, stagnant air conditions. Eliminate them and you don't get the molds and fungi.

Dave