The most common way to start a fire is a Sim cooking with a very low cooking skill.
If that's the case, your Sim will catch on fire the most often.
The case of fire for cooking accidents are very random but influenced by Cooking skills.
One of my single Sims who were living in a small mobile home, with a standing foundation, caught on fire from cooking, however he was able to "Extinguish Self" and survived. But he was singed after, as sooty and black. Using a very cheap stove like the YumCooker helps with this.
Although it is believed that a rocket can catch fire, this is false.
Sometimes the failed fireball trick will cause a fire on the bar and catch the Sim on fire. You just have to be lucky for this to happen.
A child Sim can catch on fire for using the Junior Wizard Science table, or any science table they used for too long. This could catch on fire.
For Outdoor Retreat, there is a chance that the objects that are used near a campfire will catch on fire, and the Sim catch on fire as well. Once again, there is no way and the only way to fix this is to move away.
In GTW, using the transform ray may light objects on fire and a problem using the Invention Constructor may cause a fire. This is all on luck.
Also, keep eating Rose Perfume serums. This will help too, and could catch the Sim on fire.
Spa Day has an incense holder. This, to me is probably the shortest and efficient method to use. Use a wooden table, light the incense, and use a rug underneath the table or wooden floors. This WILL catch a fire!
Launching fireworks indoors with wooden objects all around. this will likely cause a fire. Only for City Living.
Sometimes, using a DJ booth for a long time will cause a fire. You can prevent this by simply adding the "Advanced Heat Sink" option.
If your adventurous Sim is heading for the Omiscan Temple, using the wrong switch may put the Sim on fire. Otherwise, other negative actions.
For Snowy Escape, using the Best Friend Forever space heater for a long time can cause fires. Wooden floors have a lower fire resistance and it can be prevented using the Fire Safe upgrade.
Similar to the past versions, using a rug is not advised near a fireplace because it could catch on fire(?) And using the Fireguard can help with this(?)
In Eco Lifestyle, using the "Composting Container" and not emptying the waste will catch the toilet on fire.
Also, leaving a power generator on for too long could catch a sudden fire. This happened to me once.
That's all the ways to catch on fire and Kill your sim.(As I remember) Enjoy!