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Nick_Tarleton comments on Shock Level 5: Big Worlds and Modal Realism - Less Wrong
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IANAPhysicist, but I'm fairly sure that space and time are entirely continuous in standard QM or QFT, though they are discrete in loop quantum gravity and possibly other theories of QG.
Standard QFT doesn't have discrete space and QCD may make sense with continuum of space-time, but models with a Landau pole, like QED and the standard model, don't make sense at small length scales. The length at which the Landau pole appears in QED is smaller than the Planck length, so no one cares about it, since they expect bad things to happen already at the Planck scale.