No output reading from Op Amp on Oscilloscope
There are two problems that could be causing grief, the first is the mode of the oscilloscope, the second is the ground of the oscilloscope. 

The first thing to do is validate your equipment, which you already did kind of. Make sure the oscilloscope is in DC mode, and measure a DC voltage. The probe may be capacitance coupled (in AC mode) which would explain that you can measure a 5V 1kHz square wave but not a DC signal. 

Measure a battery or power supply rail to make sure the oscilloscope is measuring correctly. Once you have measured DC signals then return to the op amp and make sure your oscilloscope ground is the same of the op amp.

Edit: 
On most scopes the ground is connected to mains ground and not isolated, so if you connect a ground point to something that isn't ground (or a virtual ground that isn't isolated) then a short to earth ground will be created), check the point that you are grounding and make sure it's really ground. Another way to do this is to measure the voltage between the point you are connecting it to and the oscilloscope ground with a DMM. If it measures something more than a few 10's of mV you may have a problem.