Sadly, Humphrey Bogart has probably made it almost impossible to wear a trenchcoat and a trilby at the same time and not look as if you're on your way to a costume party. However, the trenchcoat remains a very practical bit of clothing and, if you're ever caught in a heavy rainstorm, something you'll be incredibly glad to have with you.
Acquascutum and Burberry both claim to have invented the trenchcoat, some time in the middle of the 19th century but, as the name suggests, they were developed and popularised during the trench warfare of WWI. Their military use gave them the shoulder-straps and D-rings that they still have today while their restriction to officers meant that, in the post-war era, they developed an association with respectability that made them appropriate for business-wear.
More importantly, if you do get caught in the rain, they're brilliantly practical, with a high collar that can button up to protect your neck and face, an inbuilt cape to help keep your back and shoulders dry and, generally, a removable liner for extra warmth. If you live anywhere it's likely to rain, a decent trenchcoat will compliment your suits much better than the tatty windproofs that a lot of men wear.