I shaved mine but it was because the clumps of hair falling distressed me and I felt better just cutting it off. I had brushed and brushed ...washed my hair with great clumps coming out and then brushed and dried...only to find more coming out. I felt relieved to get rid of it.
You can pop a cap on to collect falling hair but it may matt. Have you organised a wig. Our local hairdresser here does this...she matches whats best and will shave then cut your wig to suit you. Your breast care nurse should have info about this as should your chemo unit.