What does Honda call for? Regular, right? The compression isn't high enough for them to call for anything better. Thats what you use. Using a higher octane will not improve economy or power and may cause drivability problems.
Like the above poster noted, higher octane (Premium) gas is generally specified for higher compression engines where the additional octane helps control detonation, knocks, and pings in the combustion cylinders. Octane is a measure of the fuel's rate of burn, the higher the octane rating the slower the fuel burns. Higher octane is necessary for high performance applications but it comes with a financial and a mechanical (combustion) cost. Basically, using higher octane in an unnecessary application will actually cost you performance, not gain it. Albeit, very slight and probably undetectable, theoretically your mileage should suffer by using higher octane gasoline.
Octane and it's effects are some of the most misunderstood and misrepresented topics in motoring.
The octane is simply a measure of the level to which the fuel can be compressed before it spontaneously ignites, or 'diesels'.
As your Honda has no way of miraculously raising it's compression ratio, the engine has no means of utilising this extra compression-capacity, so will gain no benefit at all. The only effect you will experience is your wallet emptying slightly faster.
If you want to spend more money on your fuel go for a high quality brand at the octane given in your owners manual.
Higher octane fuel can be of benefit to deliver a higher degree of spark advance without detonation which may increase performance & also may allow the engine to be run leaner on cruise before detonation occurs. Typically a factory mapped ECU won't take advantage of this, which means you're just wasting money on fuel you can't take advantage of. I have, however, seen cars gain performance & economy from running higher octane fuel after being remapped.
Stick with the recommended octane rating of your vehicle. Octane rating is a fuel's ability to resist pre-detonation. That is on the compression stroke of a gasoline engine, the fuel/air is injected into the cylinder and crushed. At a set point the spark plug is fired and ignites the mixture. If your engine required 93 octane and you put 87 octane in the engine, there is a chance that the fuel/air mixture will ignite before the spark plug has a chance to fire. This causes pinging and while not normally bad for the engine if it happens once in a while, it can cause problems if it happens frequently.
Please note that no current vehicle on the road can detect the octane rating of fuel. Those are measured using the Motor and Research Methods (R+M)/2 that you will see on the gas pump. It is an average of the fuel. The engine can only detect problems via knock sensors which listens for pinging. If a knock sensor is triggered it will cause the car to go into limp/safe mode. This will pull timing on the engine, it will pull power and probably prevent you from going over a certain RPM in order to protect the engine (Newer Vehicles). Obviously this is only in a severe instance and would require a visit to the dealer or mechanic. In the end just stick to the owners manual and learn to drive your vehicle more efficiently in order to save on fuel, money and piece of mind.
Car's can detect the octane of fuel used, indirectly. It called the ping (knock) detector. If an engine has a particular CR (compression ratio) by advancing timing, till it pings, which is why the ping detector is there, the cars computer could calculate the octane that is used. By advancing the engine you are increasing the CR. It's common way to increase HP, by increasing the timing.
The bottom line, rarely using higher octane, is a cost effective way to increasing fuel economy.