How do Sources Affect the Linearity of a Circuit

How do voltage/current sources – both dependent and independent – affect the linearity of a circuit?

To apply the superposition theorem, a circuit must be linear. As I understand it, this means that the circuit must contain only linear elements. I understand how resistors, capacitors and inductors meet this requirement.

I don’t understand how the linearity of a circuit is affected by sources. I don’t think sources can necessarily be linear elements.

Let’s say we have a 1 volt voltage source attached to a 100 ohm resistor. Now let’s add a 0 volt voltage source in series. This voltage source has 0 volts of voltage and .01 amperes of current across it. A linear component must obey the homogeneity property. An input of 0 must be matched by an output of 0. This is clearly not the case for this particular voltage source.

How can a linear circuit contain – seemingly – non-linear elements?