Z80 control pins not reading a defined logic level
You should pull the inputs /INT, /NMI, /BUSRQ, /WAIT and /RESET high with resistors. 

/HALT is actually an output (including it in the list of inputs was my mistake) so it should be either high (usually) or low (only when the CPU is executing a HALT instruction).

Here is an example test circuit for the Z80. It connects all input lines except /RESET directly to +5V, which is OK if you don't ever want to use them (just make sure you connect the correct pins, as an output pin might not like being shorted to +5V!). When the CPU is running The LEDs should blink in a binary sequence, showing that the address is continually incrementing.