Carrying things on a bicycle is more of an art than a science. If you've got a bakfiets, you just throw stuff in there, whether it's groceries, kids, potting soil, anything short of a piano is cool. However, that kind of pedal-powered wheelbarrow is pricey, and the rest of us have to make do with plain old bikes.
Common cargo for most of us may be a laptop or a couple groceries, but I also find myself hauling musical instruments to practices or, occasionally to performances. I've come up with a couple of ways to pull it off.
If the instrument is small enough and has a case with straps, you can, of course, just sling it over your shoulder, but in the case of many string instruments the neck tends to stick up awkwardly and bang me in the back of the helmet as I ride.
You can also try...
On a rack
Hauling instruments on a non-cargo specific bike is great for smaller items, such as my mandolin, pictured above. You can simply strap it to a rack and go. This also works best if you have a hard or semi-rigid case with a fairly regular shape. My old mandolin case was shaped like a mandolin (basically a tennis racket shape) and ended up with all the weight on one side, which made it hard to attach securely. My current case is rectangular and ties down nicely.
I also once saw a guitar mounted on one of those side racks used for surfboards, which was modified to hold a lightweight guitar case. I haven't seen a commercial version of a bike-mounted guitar carrier yet, but that doesn't mean there isn't one out there.
If you play a smaller instrument, you can also just drop it in a basket. I'd use a bungee cord or two to make sure it doesn't bounce around too much, but it's a good solution. If you have a larger instrument, or need some stands and sound gear as well, you can try...
On a trailer
The trailer itself is a just a child carrier with the seats and sides stripped off, leaving it a simple flatbed. It works fairly well, and doesn't affect the handling of the bicycle all that much, even with a good amount of weight on board (there is a bit of a difference in using the brakes while rolling downhill). If I'm carrying my guitar or a similarly long-necked instrument, I just put the body on the trailer and let the neck of the instrument hang off the back of the trailer (in a hardshell case). Somebody once asked me if I worried about the road vibrations doing some sort of harm to my gear, but I explained that, being acoustic instruments, they're made to vibrate, since that's how they work. Maybe if I had more delicate electronic stuff, I'd worry, but basic guitars and mandolins are fine.