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Some of the particular herbicides that work best to get rid of nut grass without killing the grass include Halosulfuron-methyl, mesotrione, sulfentrazone, and metsulfuron. A herbicide like glyphosate would kill nutgrass, but it will kill nearby plants, too.

Weeds can grow through rubber mulch, but it depends on if you removed all the weeds in your garden beforehand. Weeds already in the soil can grow even when using rubber mulch. However, that doesn’t mean that your garden will be littered with rubber and weeds, but it does mean that the occasional weed might still pop up. Another way you can increase your weed suppression is by using landscape fabric as a second layer. 

They compete for nutrients and water, potentially stunting desired plants.

No, definitely not. Pulling weeds by hand is the single most effective way to remove them because you're eliminating the entire plant—roots and all—so it won’t have a chance to grow or spread. Setting time aside a few times per week to pull weeds can keep them at bay. If the number of weeds gets overwhelming, then you can resort to other methods.