I have a single handled kitchen faucet that brownish water comes out of. It makes no difference if it is hot or cold, the brown water will flow for 30 - 90 seconds. Sometimes it seems to be clear and then the next time it will be brown again. This is the only faucet in the house that I am having this problem with

Somewhere in your water lines leading to this faucet you have some iron pipe, I suspect that it may be real close to the faucet itself. The water washes the rust out and the water clears until it is off for a while and therefore the water get brown again.