So this isn't a base of a program, but hopefully it'll get you started with DirectSound.
If you want to use DirectX in a Managed program, the supported method is using XNA. The current version of XNA can be downloaded here:
http://create.msdn.com/en-us/home/getting_started[
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Don't worry that it talks about Windows Phone and the XBox 360, it works on PCs as well. This page will then help you get started with recording sound:
http://msdn.microsoft.com/en-us/library/ff827801.aspx[
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You then need to work the processing magic on the data you get. Walt's recommendation of looking at DSP libraries is a good plan. I don't know much about this sort of thing, but you may want to look at Fast Fourier Transforms, which will let you extract frequency data from a signal. This may help you work out what pitch your singer is singing at. Naturally, this needs to be done as fast as possible to give your user timely feedback.
Hope this is of help. Good luck :)