Why
Wheat noodles, somyeon and pasta aren't toxic, but they're mostly carbohydrate with little nutrition for pets. Long strands slurped down can cause gagging or choking. The real problem is usually the broth or sauce: noodle soup stock, jjajangmyeon and tomato or cream pasta sauces contain salt, onion and garlic. Some pets are allergic to wheat.
How to give it / if eaten
If you want to share, rinse plain, unsalted boiled noodles and cut them short: no more than a chopstick-pinch for cats and small dogs and a few pinches for large dogs. Don't give ramyeon, jjajangmyeon, jjamppong or pasta with sauce.
Watch for these signs
Too much may cause loose stool or gas. Long strands swallowed whole can cause gagging or vomiting. After seasoned noodles, watch for vomiting, diarrhea, extreme thirst, and over the next days weakness or red urine (onion/garlic poisoning).
FAQ
My dog ate a few strands of jjajangmyeon.
Jjajang sauce is full of onion. A few strands can often be watched, but call your vet if a lot of sauce was eaten or your pet is small or a cat.
Are rice noodles or buckwheat noodles different?
The noodles themselves are similarly okay in small amounts, but the broth and sauce are still the issue. Buckwheat can rarely cause allergies.
General information only. Weight, age and health conditions change how a pet reacts, so ask your vet if you are worried.