When I try to make small dots on my cookie decorations, sometimes the royal or syrup icing gets stringy so it leaves a trail instead of coming off the dot clean.
As Escoce mentioned, it's partially technique.
After you finish your dot, you want to move the tip in a quick circle while keeping the tip against the dot, then lift off. This will help to minimize the peak created.
If it's only a peak, and hasn't put icing all over the place, you can take a damp brush and knock down the peak.
'Threads' might be too fine for this, but typically if icing has gotten where you don't want it (eg, typos in words), it's often better to wait for it to harden, gently break it free with a toothpick, and then brush it away with a soft brush. If you really mar the surface in the process, cover it up with a flower or other decoration.
I had this problem piping rosettes. It came out stringy.
I think the icing might have been too soft. I put it back in the fridge in the bag, for a few mins to firm up. Or take it out of bag and remix it.
If that doesn't work, there might be air bubbles in it. Take most of it out of the bag and remix it. Take your small spatula and remove the mess.
But, in my case, I firmed it up in fridge and it worked.
Also, having a cold cake or cupcake makes it easy to clean up mistakes.