What single word describes the action of tapping a spoon against a glass?
What is it called when someone taps a glass with a spoon in order to seek attention at a party or at the dining table?
A single word for making a sound by gently hitting a metal object against a glass one could be clink:
If two glass or metal objects clink, or if you clink them, they make a short ringing sound when they are hit together: Spoons clinked against the crockery.
(Longman)
Here's a real-world usage example from a book called "Mama's Little Book of Tricks":
(Source)
If a phrase is OK, you already mentioned it: it's tapping a spoon/fork/etc. against the glass.
Consider, tinkling a glass.
TINKLE
: to give forth and make a succession of short, ringing sounds, as a small bell
: to make known, call attention to, attract or summon by tinkling Random House
: make or cause to make a light, clear ringing sound: [no object]: cool water tinkled in the stone fountains; [with object]: the maid tinkled a bell Oxford Dictionaries
: to make or emit a tinkle or a sound suggestive of a tinkle M-W
: to make light metallic sounds, as those of a small bell: v.tr to cause to tinkle; to signal or call by tinkling. American Heritage® Dictionary
He returned to the sitting room and turned down the music, while Guy tinkled a glass to gain everyone's attention; The sound of a spoon tinkling against a glass again brought a hush to conversations.
A Year at Nethercombe Ley
Hal tinkled a glass to get everyone's attention. "I've got some hot news for you guys."
The Happy Accident