The best time to control weeds in asparagus is early spring before the asparagus emerges. A light tilling (or hoeing) that is shallow enough to avoid the crowns will eliminate existing weeds. Many gardeners like to mix in organic matter during the same operation. To keep weeds out, apply mulch.
Herbicides can also be used before asparagus emerges. Glyphosate (Roundup, Killzall) will kill weeds that are actively growing, and the preemergence herbicide trifluralin can be used to kill weed seeds as they germinate. Trifluralin is found in several products but not all of them list asparagus on the label. Those that do include Miracle-Gro Weed Preventer Granules and Monterey Vegetable and Ornamental Weeder.
No herbicides can be used during harvest. However, the end of harvest presents another opportunity. Remove all fern and spears, lightly till the bed, and mulch. You may also apply Roundup after the top growth has been removed to control virtually all of the weeds present.
Past the harvest season and after regrowth of the asparagus, options are limited. Products that contain sethoxydim can be applied to asparagus to kill grassy weeds. Sethoxydim has no effect on broadleaves. The two sethoxydim products I’ve found available to homeowners that are labeled for asparagus are Monterey Grass Getter and Hi-Yield Grass Killer. So what about broadleaves? Really, your only option is to pull them and look forward to next year.