I've always had problems with root vegetables because of shale & clay in my yard. Here's an article that talks about about root depths. Unfortunately, the article didn't help me with my garden woes involving root vegetables. I'm amazed at how far down some of those tap roots grow.

Here is a link that might be useful: Vegetable Root Development

My property is all mud shale so the only way to go was to dig out and amend up. Most radishes except the daikon types have grown fine for me in 12 inches of clear soil. Potatoes were fine as well at that depth since none of them were growing below where I place the seed piece and I hill up. I've only done long carrots once (usually grow thumbellina) but I don't remember them coming out really crooked at 12" of good soil. The roots definitely go down farther the production is usually higher up.

I think 12 inches is more than enough. I haven't grown carrots but radishes and potatoes grow fine in my soil that is only 8 inches deep in some places before I start getting to the rocks. If you are moving a lot of the soil and don't mind a little extra work I think your garden would really appreciate loosening of the subsoil. Just move the top 8 or so inches of soil to the side and stick a pitch fork into the subsoil and just rock it back and forth. You don't need to lift or turn it, just loosen it. In my Maryland garden the soil has a good deal of clay and can remain damp at times, loosening the subsoil can help with drainage.

Thanks for the info folks. I'm at around 18 inches so I guess I'm good.