Any time you use electricity you have the risk of a short or fire. It doesn't matter what the heating device is, you need to be careful of the basic electricity. That's the wiring getting the electricity there and the connections and fixtures. That's just basic.
To me a brooder needs to have one spot warm enough in the coolest temperatures and a spot cool enough in the warmest temperatures. If you are brooding in a climate controlled area where the temperature is stable that's not that hard to set up. My brooder is in the coop. The ambient temperature can swing from below freezing to the 70's Fahrenheit (over 21 C) in a few hours so my brooder heating system needs to accommodate that. I find that the chicks are really good at regulating their own temperature if given the option, even straight out of the incubator.
I use a heat lamp. I get rid of that clamp that comes with it, I just don't trust it. If you use a heat lamp, use wire or chain to securely hold it in place. To me, that eliminates the major fire risk. It can't be knocked down, by you or a chicken. With a heat lamp make sure it is secured. If you don't do that I do consider them a fire hazard.
My brooder is fairly large and has decent ventilation. In cold weather I keep one end toasty and let the other end cool off as it will. Sometimes there is ice on that end. In summer I use a lower wattage bulb, they don't need that much heat.
Heat lamps are often not the best to use on smaller bins or aquariums. It is easy to overheat something like that. It needs to be big enough so that one end is warm while the other end is cooler.
Many people like to use a heating pad. I don't use them myself but Blooie has a long thread on here about using them. She uses them outside in freezing weather, they work. There are some tricks to setting them up so it would be a good idea to find her thread and read through some of it, it is a long popular thread. Her husband was a fire fighter and mentioned that they can be a fire risk. If you use an old heating pad the wiring can become brittle and short out, causing a fire. If the plastic is brittle don't use it. Also, some come with a timer to shut them off after a couple of hours. You don't want that kind of safety feature. As long as they are in good shape and you set them up properly I consider these very safe.
I don't know of any specific fire hazard associated with a heat plate. Some are only rated down to a certain temperature, 50 F (10 C) if I remember right. If you are in cooler temperatures you might want to check the manufacturers recommendations. And check on how you are supposed to adjust it. There are always tricks with whatever you use. If you use them correctly they usually work great.
A few years ago a lady who is no longer active on here did a comparison with a heat lamp and one of the others, I can't remember which. She considered the heat lamp to work better with larger numbers of chicks, I think she was in the 20 chick range. You might want to watch the capacity as far as numbers with the others.
Look at how big your heat plate or heating pad are compared with the brooder size. Some can fill up a brooder. To me that means make sure your brooder is big enough, no matter what you use for heat. Those chicks grow pretty fast.
To me it is more than just fire hazard. Some work better in certain circumstances than others. I personally would not use a heat lamp inside a climate controlled area. It generates heat you don't need in your house, let alone the risk of overheating a smaller brooder. Any of them can work well outside.