From a viewpoint of Health and Safety, workplaces should ban high heels outright. The proven risk of long-term damage to an employees' health and the greater risk of accidental injury make it a duty of care issue.
There is evidence that wearing a tie can damage the larynx and starve the brain of oxygen, so there is a school of thought that men shouldn't be made to wear them. I never wear one ,but most of the male managers I've had did. But given that, in my experience, most managers have no brains and talk out of their arses, I don't suppose it matters!