One of the big differences between General Relativity (GR) and Quantum Mechanics (QM) is how they each view spacetime. GR sees spacetime as a smooth continuum, like a ramp, and QM sees spacetime as discrete quantities, like a staircase.
Scientists have long been trying to unify GR and QM into one theory, and most have started under the assumption that it's GR that must give. That means trying to find a quantum theory of gravity. If spacetime itself is split into tiny, discrete units that could not be subdivided further, it would be an indication that scientists are on the right track.
Boffins at Fermilab set up an experiment called the Holometer to try and detect spacetime fluctuations at the so-called Planck length, which is thought to be the smallest possible unit about which our current physical laws can make meaningful statements. If they could detect these fluctuations it would point to space being quantised.
The Planck length is astonishingly small. The width of a human hair (about 0.1mm) is roughly halfway between the size of the observable universe and the Planck length.
A few years ago they tried looking for back and forth fluctuations and found none. They have recently tried looking for rotational fluctuations and still found none. Spacetime, as far as experiment can currently detect, is not quantum; it's a smooth continuum.
This does not mean the end of the search for a quantum theory of spacetime, though. Perhaps there's something at a smaller scale than the Planck length, or perhaps the scientists are missing something crucial but as yet unknown. "Unknown unknowns," as Donald Rumsfeld might say.