Among the less attractive but still enduring images of my youth is one of the late comedian Mr. Jackie Gleason dressed in a white undershirt with braces over the top. Though they are technically underwear, this combination probably represents the most extreme example of when not to wear braces. Do not let it influence you too much however, for under the proper circumstances braces are undoubtedly the best way to support a man's trousers.
As a wiser man than I once said, "Trousers should hang from the shoulders, not from the waist." That is because braces have the highly desireable attributes of helping one's trousers to hang better, wear more comfortably and require less adjustment over the course of the day than alternative methods. The negative is that they are more visibly obtrusive than belts or unadorned trouser waistbands, and so the rule of thumb is that they are best worn under a jacket, and best avoided without one. And because of that, they are increasingly desireable as a man's clothes increase in formality.
Another way of repeating that guideline is that braces relate to the formality of a man's clothing in inverse proportion to the amount of time he will have his jacket off. That makes them a blessing for white and black tie, where the jacket is never removed, and nearly as useful for suits, particularly vested suits. Conversely, they should not generally be worn with naught but a shirt and trousers.
It is between those two extremes that questions most often arise about brace wearing, and again I refer the reader to that guideline about jacket wearing. Personally, I wear braces under odd jackets, as does Michael Alden, proprietor of The London Lounge, in the photo, and I do so with and without a necktie (they obviously work over a knit worn tucked into the trousers, and perform the same function under a knit that is worn untucked). On the other hand, I eschew braces with shirt jackets because I tend to remove those throughout the course of the day.
And so, dear reader, let not Mr. Gleason's poor example inhibit you from wearing braces whenever appropriate.