In general, any time a contestant manages to go for a full minute without being interrupted. Kenneth Williams managed this a staggering 66 times in 337 appearances.
Bonus points (sometimes actually handed out by Nicholas) if they actually go for more than a minute without being interrupted.
Clement Freud managed three "perfect minutes" in the 14 March 1981 episode, on "after dinner speaking"note not coincidentally, something Clement was often engaged to do, "how to make a fortune", and "the best putdown I ever heard" - in the last case, going for 70 seconds. (For extra awesome, the episode also featured a "perfect minute" from Kenneth Williams on "hauntings".)
In an episode from August 2000, there were three "perfect minutes" in a row, managed by Sue Perkins (on "bath time"), Jeremy Hardy (on "bringing home the bacon"), and Tony Hawks (on "jumping to conclusions").
Sue Perkins managed this twice in one episode in September 2013, on "Marcel Marceau" and "last orders".
On more than one occasion, Paul Merton has spoken for beyond the entire minute because nobody wanted to interrupt him.
From the 2012 TV series, given the subject "Pardon my French", Gyles Brandreth, Tony Hawks and Nicholas Parsons all start playing the game in fluent French for the first half of the minute, much to Paul Merton's bewilderment.note Worth noting that, as a boy, Gyles was a student for several years at the Lycée Français de Londres (now the Lycée Français Charles de Gaulle), one of Britain's elite French language schools.