I don't know who keeps giving your questions the thumbs-down - someone with an ax to grind? Hmmm.
I've only ever used charcoal grills, and my safety rules are as such:
* Keep everything on a concrete surface, if possible. I used to live in a place with a wood deck, so I had to keep the grill there, but I lit the charcoal chimney on the concrete walkway and made extra sure to put out any little bits of charcoal that fell out of the chimney when I was taking it to the grill. I may have been taking a foolish risk in doing this.
* Make sure nothing is close to the fire. Keep the grill out in the open, although I've never sensed any real danger from moving under the patio roof if it was raining or snowing. (I don't recognize the notion that grilling is just for summer.) The ceiling is high enough that the heat is really dissipated by the time it gets up there, plus there's almost always a gentle breeze to spread it out even further. But don't place the grill next to anything wooden, plastic, metal, or otherwise capable of either catching fire or becoming melted, damaged, or hot enough to pose a not-so-visible hazard.
* Keep a fire extinguisher handy just in case. Also a spray water bottle for flare-ups, which shouldn't be that much of a danger. But we are talking about open flames here.
* If you have small children, erect a barrier and make sure you know where they are at all times. Try not to leave the grill unattended. Grills are a good teaching tool for fire safety when the kids are big enough to learn about it.
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* If your grilling area is close to the yard, keep it watered during dry spells so you won't have any danger of a backyard brush fire.
That last bit is nearly impossible if you're doing some real slow cooking (barbecue, roasts, etc) because that can take an hour, or several hours. So having the grill in an out-of-the-way location is advisable if it's not impractical. But be sure to check on it frequently.
I don't know how much of this is a consideration for gas grills, but I do know that cheaper grills, even from name brands, seem prone to the lines breaking at the connections. I saw one once that was flaming up right at the control knobs, and have heard that this isn't uncommon. If I were to get a gas grill, I would try to research which models were well built and had a reputation for lasting, because that flame-up I saw was far and away the most hazardous thing I've ever seen at a cookout.