Unless you live in Antarctica or the Sahara desert, Just run 10w-40. Most import motorcycles specify a 10w-40 motor oil (and you'll find that the Honda manual doesn't specify to run a "motorcycle specific" oil). Motorcycle oils are a waste of money, IMHO. The API ratings of motorcycle oils are very dated (SG,SH ratings while most modern "car" oils are now at SN), and they charge you at least 50% more for motorcycle specific oils no matter what the case. You will have no issues running a "car" 10w-40 oil, so long as the API service label on the back doesn't say energy conserving/resource conserving.
Look at any off the shelf motor oil with a second number less than 40 (Example: 10w-30, 5w-20) and you'll almost certainly find that it is an energy conserving oil/resource conserving oil. Energy conserving oils have friction modifiers (like molybdenum or graphite) that protect metal to metal contact when the oil film fails in extreme situations. The downside to these modifiers is that they contaminate the clutch and cause it to slip (a bit too "slippery"). So as a rule, avoid energy conserving oils. I once put an energy conserving 10w-30 in my Honda 100 dirtbike (before I knew) and the clutch began slipping after the first ride. The next time I tried to start that bike, the clutch slipped every time I kicked it over, and the crank wouldn't pass the compression stroke. Motor oils (be they "car" or "motorcycle" specific) that have a second number 40 and above (Example: 10w-40, 15w-40, 20w-50) will almost never be marked energy conserving. I have never found one that was, but they may exist.
As far as synthetic goes, the same rules apply. Don't use an energy/resource conserving oil. Make sure that it is the right weight (10w-40). Pour it in your bike and enjoy synthetic bliss.
Synthetics are better than conventional oils. They have a higher temperature tolerance and develop much less sludge when used longer than the interval service is specified, which, for our bikes, is about 1500 miles per oil change. They shift smoother when cold and have less false neutrals when hot - this is my experience. Results may vary. But you may not even notice the difference in synthetic other than by the weight of your wallet... as synthetic is always more expensive. I use it for the peace of mind when I'm in rush our traffic on a 100 degree day - stop and go, stop and go.
If you love your bike and want only the best, buy a synthetic. To save money, buy a "car" synthetic off the shelf. Just make sure it isn't energy/resource conserving. If you wan't to save money, even the cheapest 10w-40 oil will do. Go to Wal-Mart and just buy the house brand conventional 10w-40 and enjoy your $10 oil change. There is nothing wrong with SuperTech oil - Used Oil Analysis proves that it is just as good or better than most name brand oils (when used in cars). And it is dirt cheap. If you are frugal, but still want the best oil you can buy motorcycle specific synthetic Valvoline 10w-40 oil for about $8 a quart at Walmart. I have yet to come accross a better deal than that for a motorcycle specific synthetic oil. You can pretty much bet that Walmart prices are better than anyone else's for motor oil, unless someone's having a sale. (Flame on haters)
As for me, I use car oils.
I buy Mobil 1 synthetic 10w-40 "car" oil. It works great in my '75 CB550. And it only costs about $6.50 a quart whereas most motorcycle specific synthetics cost $10.
When I'm running low on cash, I buy Shell Rotella 15w-40. I only run this in the summer as it is thicker at startup than 10w-40, which causes it to take a few milliseconds longer to start lubricating the top end. Rotella is one of the few "car" oils that is actually recommended by the manufacturer to be used in motorcycles. That's what sold me on it. Shell doesn't even sell a motorcycle specific oil, IIRC. Rotella is $13/gallon at Walmart. Change the oil, use 3 quarts. Have an extra quart around to top it off before a ride.
The choice is yours.
-Austin
