From Ground: First, water the common mare's-tail plant lightly to dampen the soil and make it easier to work with. Then, using a spade, gently loosen the soil around the plant, careful to not damage the roots. Begin to dig a trench around the plant, keeping a distance to ensure the root ball is not damaged. Once you have dug deep enough and around the plant, you should be able to gently lift it from its original location.
From Pot: Water the common mare's-tail plant and wait for a few minutes. Hold the plant at the base and gently tilt the pot sideways, tapping it to loosen the soil. Continue to gently pull and tap until you can slide out the whole plant along with the root ball.
From Seedling Tray: Water the common mare's-tail seedlings and let it seep through. Use a small spade or a suitable tool to gently scoop out each seedling, making sure you keep the roots intact.