I just bought a fan from Walmart and actually read the instructions. It gave one warning telling not use it with a solid state speed control device. What is that?
A solid state speed control contains components such as triacs which are semiconductor devices with no moving parts which can control the flow of current to a device. They are useful because they can be controlled quite precisely to turn on and off at specific parts of a waveform. One problem with running a fan off one is the inductance of the fan or the capacitance of the start winding causes issues with accurately controlling the shut off and start time of the triac. The choppy waveform that the triac produces can cause the windings of the motor to heat up more than usual. I’m guessing that’s why they advise against using yours with a solid-state control. The winding insulation on that motor is probably very cheaply done and can’t take the extra stress.
In simpler words, don’t plug the fan into an outlet controlled by a dimmer switch.
Or don’t push a stick into the blades. Or your finger. 
If I hook a solid state relay with a snubber circuit installed, do you think I could still run this same fan without a problem?
I’ve seen this warning on energy saving bulbs (But it says ‘dimmer switch’ rather than ‘solid state speed control device’)
[Ethel] “Bert. It’s too bright in here!”
[Bert] “Just interface with the solid state speed control device on the wall there”
I don’t think so but I’m out of my element there. I fix stuff rather than design it. Someone like beowulff or eng_comp_geek may be be able to advise on that.
You might want to check if it’s against code in your area before trying anything anyway. It’s not permitted here (Canada).
Its for a school project. I’m essentially trying to take a microcontroller (Basic Stamp 2 from Parallax.com), send a small current through the SSR to turn a fan motor on. I read in the documentation that a Solid state speed controller shouldn’t be connected to it to avoid fire? lol, but I also have documentation from my SSR that says I could, assuming I use a snubber circuit (or a metallized polyester film capacitor in series with a resistor). This is of course assuming its a light inductive load. I’m assuming this little 110 V fan motor is a light inductive load.
Oh, that’s a different story. I think you’ll be safe codewise. 
Yeah, the warning is because something like an exhaust fan could overheat and start a fire. For a project I think you’ll be fine. The worst thing for those little motors is trying to run them too slow. The rotor doesn’t spin fast enough to generate enough counter-emf to keep the running current low. If you’re going to run the motor at a lower than normal speed for very long you might want to have a spare on hand.
The problem is that the dimmer circuitry will be damaged by inductive kickback. Normal light dimmers are not suitable for controlling inductive loads like fans and motors due to the current surges that can happen with a sudden change of the load (such as switching the fan between speed settings). What is worse, if the triac fails in such a way that it becomes a half wave rectifier, it will probably take the load out as well resulting in lots of smoke and maybe a fire.
There is also a phase (timing) problem in inductive circuits where the current tends to lag behind the voltage. More advanced dimmers have compensating circuitry inside to allow installation on inductive loads.
Remember you do not want to let the smoke out, that is a bad thing.
How big is this fan and what is the leakage current in your solid state relay?
A mechanical relay is either 100 percent off or 100 percent on. There is no in between.
A solid state relay is never 100 percent on or 100 percent off. It’s always somewhere in between. If there is too much leakage current when it’s off your fan might spin a bit even when it is supposed to be off.
You could always take the output from the basic stamp through a transistor and use that to drive a mechanical relay. Make sure you put a reverse biased diode across the relay contacts or the energy stored in the relay coil could reset or fry your basic stamp when the relay de-energizes.