Room temperature: Freshly cut pasta can be stored on a semolina-dusted baking sheet for up to 2 hours. I recommend covering it with a kitchen towel — and if it begins drying out, you can wet the towel to give it a bit more moisture.
Fridge: Fresh pasta can be stored in the fridge for 24 hours. Technically, it’s fine up to 48 hours but it will start taking on a gray-ish color from oxidation. To prevent stickage, line an airtight container with parchment paper dusted with semolina flour. Place the nests of pasta on top, then cover with more flour and another piece of parchment paper. Seal completely. If you have a lot of pasta, I recommend storing it in an even layer on a wide tray that can be covered.
Freezer: For longer-term storage, the freezer is a great option. Layer the nests of pasta in an even layer on a parchment-lined baking sheet. Freeze for 1 hour until solid, then transfer to an airtight container or ziploc bag and freeze for up to 1 month. Cook straight from frozen.