I hate hangnails. Haaate them. I’ve been carrying cuticle nippers in my bag since college and my favorite are these mini cuticle nippers made by Tweezerman. They are slim, lightweight and I’ve never been asked to remove them from my luggage when I fly. They come in a small kit of tools but the nippers themselves have a tight fitting rubber cap so you can just slip them in your bag and won’t have to worry about anything being scratched.
Tweezerman also makes two slim larger sized nippers: the Squeeze-n-Snip, which has pink accents, and the Squeeze + Snip Ultra, which is stainless steel. These larger ones are about 4 inches long where the mini ones are 3 inches, about the size of tweezers, and are quite a bit lighter weight.