Some single supply op-amps have a common-mode range that extends slightly below the negative supply rail.
At least one TI op-amp has a built-in charge pump and the common mode range extends 200mV below the negative supply rail and 200mV above the positive supply rail. So do some R-R input op-amps (at the expense of a "kink" in the input response- they're really two front ends that have to pass off at some input common mode voltage).
But generally for the most selection and best price/performance in op-amps you wan to keep both inputs within the supply rails, and often well within the supply rails (no closer than a volt or two for some op-amps).
Input common mode range (or sometimes Input Voltage Range) is the data sheet spec you need to look at.
For example, the OP-07 has a specified range with +/-15V supplies that is +/-13V.
The LM324 has a range from 0V to 2V below the positive supply (depending a bit on temperature range etc.) and is extremely cheap.
The MCP6001 works with inputs as much as 300mV outside the power supply rails (and is fairly cheap).