If you search the site, you'll find the issues well documented, but I'll give you a short list to do your searches on.
First, and most importantly, make sure your dealer checks it over and ensures all the updates and recall work has been done on it. That addresses some of the complaints you will find here in the forums.
The list (others may chime in with more)
DPS(Power steering) : A couple of recalls on this, and the latest was announced late in 2010. They are replacing the legacy DPS in the Spyder with a new design. Contact your dealer to confirm yours was done.
Evap Canister: The is a short hose that the dealer should add on some Spyders.
Relays: Located under the trunk hood with the front fuse panel. Tend to shake loose and need to be reseated. Should have a piece of foam on the relay cover to help keep them seated. Should be checked now and then.
Y Pipe Gaskets: These are on the exhaust pipes on either side of the Spyder. The factory gaskets have a tendency to wear out rather quickly. Some, like me have had to replace them at 6000 miles or less. Some of us are using the better quality Honda gaskets now.
Brake Failure Message: The Spyder is very sensitive to brake fluid levels. If you get this message on the front cluster, you should check your brake fluid. Though they will look full, one will be lower and require 1/2 ounce or more to make the error clear.
O2 Sensors: These aren't a common failure, but some have had to have theirs replaced. It will throw check engine light on you that will come and go.
Oil Pressure Sensor: This has failed on many Spyders. Typically, it is just the sending unit, and the spyder runs fine. Other times it has been pump related, so please ensure you properly diagnose before riding again.
Clamps: Numerous issues with hose clamps working loose. Most notably with the coolant system, and some with the fuel system. Give them a once over now and then.
Coolant Tank/overflow tank: Develop cracks over time and leak. In some cases, a failed radiator cap causes the tank to split.
Front tire inside edge wear: Happens on most every Spyder I've seen. Inside edges of front tires will wear more than outside. Seemingly normal by design. Some have gone to other brands of tire, but also found the same thing. Not a big deal, just rotate them. Many thought it was simply alignment issue, but seems to be part of the spyder's personality.
Parking Brake: There is a factory update to these for the pedal assembly, but they do go out of adjustment often and require 20 minutes to adjust for proper function. I have to do mine about every 3 to 4 thousand miles.
Battery: Some have noted the stock battery doesn't last long. Beats me, as mine is going on two years since new...
Tail lights: Are a royal pain to change. I didn't have one burn out until about 10k miles... I modified my tail light assembly to change them easier.
That is the short list. Doesn't mean anything bad about the Spyder. Not everyone has had all these things happen, but do keep your eye on them and just be aware.