Op-Amp based schmitt trigger working in simulation but not in reality
Did you use the model of the MC1458 in your simulation, or a generic (ideal) opamp? In the latter case that may explain why it doesn't work. The MC1458 is not a rail-to-rail opamp, and can't work near the rails both for input and output. Especially with a low supply voltage the operating range of the opamp is very limited, and is situated around Vcc/2. The Schmitt-trigger thresholds and your input signal are way below that.  

Still, the output signal looks rather strange. What's the scope's vertical resolution? Anyway, I'd try a rail-to-rail opamp first, that's something which has to be fixed anyway, and see what results that gives. BTW, you could also bias the inverting input to Vcc/2 via a resistor divider and couple the signal in via a capacitor, but you'll still have a problem with the output levels. If you use the divider, don't forget to recalculate the Schmitt-trigger resistors.