Visual inspection is the best way. Dull router bits are shiny in spots along the profile cutting edge. You can see this by looking at the cutting edge as you move the bit around in the light, dull will give a reflection off the cutting edge. For the first time you might want to use a magnifying glass to take a closer look at the cutting edge. When the router bit is sharp, you won’t be able to see the reflection on the cutting edge.
If dull, they are harder to push through your material so if you are using a bit that seems a little hard to push the material through it, check it. Most of the cheaper router bits can be disposable but your more expensive router bits can be re-sharpened at a local carbide sharpening specialist.
If the router bit has a bearing, it is also a good idea to rotate the bearing between your fingers to feel it before you mount in the router. If its rough feeling it needs to be changed, if its moving smoothly then use it.
My thoughts on dull router bits,
If dull, they are harder to push through your material so if you are using a bit that seems a little hard to push the material through it, check it. Most of the cheaper router bits can be disposable but your more expensive router bits can be re-sharpened at a local carbide sharpening specialist.
If the router bit has a bearing, it is also a good idea to rotate the bearing between your fingers to feel it before you mount in the router. If its rough feeling it needs to be changed, if its moving smoothly then use it.
My thoughts on dull router bits,