Outside of the gentle clicking sound every time your system turns on and off, you should never hear anything from your furnace at all. If you do hear strange sounds (or smell strange smells) coming through your air vents during the winter, that should warrant an immediate call to your HVAC company to schedule a furnace repair service. Specifically, if you hear any of the sounds below, have it looked at before these minor issues can turn into major problems and shut your unit down completely.
A high-pitched whining sound coming out your vents usually only means one thing: your fan belt. As furnace units age, the fan belt begins to grind down and get thinner, and eventually snap if it's not replaced in time. Though your unit itself will be fine, the belt to the fan blade will need to be replaced before it starts working again. The hard part in this process is not replacing the belt, but accessing the unit in the first place, so make sure that this is a job handled by a professional.
If your air vents sound like a truck bed full of gravel, it could mean you have a few parts that are broken off and are rattling around inside your air vents. It could be something small, like a washer or a bolt that has slipped off, or it could be something major, like a piece of your fan blade that is chipped off. Reattaching bolts are easy, but if it's a fan blade, it'll need to be replaced before it completely scratches up the inside of your unit and cause it to shut down.
As mentioned above, a gentle clicking sound when your unit turns on and off is completely normal, but if you hear a continual clicking sound while your unit is supposed to be operating, that's a sign that your furnace is having a hard time turning over and staying on. Usually this is a problem with your furnace's power source; a simple furnace repair call will reattach this connection point and have your unit operational in no time.
Although most homes don't use boilers anymore, many older office buildings and apartment complexes still do, so if you hear a popping sound coming through the baseboards or the vents near your space, you should contact a technician immediately. This sound usually indicates that there's a problem with your building's heating system, such as sediment that is built up on your burner, which then causes the pressure to build up and then eventually ignite. Those little popping sounds are miniature explosions in your pipes that require a heating system repair call as soon as possible. Find a company near you like Triangle Enterprises Inc if you need furnace repairs.