OPAMP single supply behavior 

Introduction

Most OPAMPs work best when powered with a symmetrical power supply having an equal positive and a negative voltage. Depending on the type of OPAMP, the output is either able to reach all the way to the negative or positive supply rail or not. When the output is not able to reach the power supply rails, it will usually clip somewhere between 0.7V and 2 volts below the supply rail.
When deciding to power an OPAMP with a single supply (a positive voltage and reference ground), care must be taken to make sure that the OPAMP will work properly, especially when the OPAMP is not a rail to rail input/output (RRIO) type.
The inputs/output of CMOS OPAMPs (f.e. MCP6022) can swing all the way to the positive supply rail and also to ground. Often these CMOS OPAMPs have a limited power supply voltage (like max 5V or 5.5V). 
Some OPAMPs are optimized to operate from a single power supply (f.e. LM324, LM358), but that does not mean that the output of the OPAMP will follow the input all the way from ground to the positive supply rail. Typically, the output does not come closer than 0.7V to 2V from ground or the positive supply rail when the input is either 0V or the positive supply rail, except for rail to rail input/output OPAMPs.

It is crucial to first check the datasheet to find out if the OPAMP can operate from a single supply. Then check the input common mode range to see if the input can go all the way down to ground and/or to the power supply rail. Then check if the output can swing all the way down to ground and/or the power supply rail. This information will tell you how the OPAMP will behave when powered with a single supply voltage, so you have an idea what to expect when you design an OPAMP circuit.

Test setup

The OPAMPs were configured as a voltage follower powered by a single supply of +10V DC and with a 1kHz sine wave at the input having an AC amplitude of 10Vpp and a DC offset of +5V (so the sine was alternating between 0 and +10V).
This way, we can see how the OPAMP performs with an input signal that covers the whole power supply range.
None of the OPAMP are rail to rail output/input types. Some tested OPAMPs had no internal compensation but external compensation inputs. For these OPAMPs a 47pF capacitor was connected to the compensation inputs.
In the figure below, the test setup for the OPAMPs is shown:

Test results

Here follows the list of tested OPAMP and the test results:

Single OPAMPs

Dual OPAMPs

Datasheet check

First, we need to check the datasheet of the OPAMP to see if it suited for operation from a single supply voltage.
Then we need to check the input common mode range and the output range of the OPAMP.   This, we do to find out if the input common mode range includes ground and/or the power supply voltage, and also if the output can swing all the way down to ground and/or the supply voltage.

F.e. the datasheet of the LM324 mentions that the valid common mode range is from device ground to Vcc-1.5V. It also specifies the output voltage swing from rail, i.e. : how close the output can swing to either ground (or negative rail for a symmetrical supply) and the  positive supply rail.

F.e. the datasheet of the LM358 mentions that the input common mode range includes ground and that also the output can swing to ground even when operated from only a single power supply. The VOL (V output low) is defined as the swing parameter, which is typically 5mV, so very close to ground.

F.e. the datasheet of the CA3240 mentions that the wide common input voltage range typically goes down to 0.5V below the negative supply rail. So the input can go to -0.5V. The output voltage ranges from 3V for a +5V supply to 0.3V. So the output can swing relatively close to ground.