That depends almost entirely on local ordinances and, if you have one, your HOA. The far side of the spectrum where you actually have significant control over your own property is great, I'm just not sure it's very common.
All those things stack up though; if you rent a house in an HOA neighborhood you’re still accountable to the local government and the HOA, it’s just that you’re also accountable to your landlord on top of that.
I've never seen a house as restricted by HOA or local ordinances as the Internet would lead me to believe, so I'm on the side that it very common to have a lot of control over your house.
Source: used to rent, now own