Home improvement contractor here.

Short answer: it’s always inherently risky. There’s no 100% certain way to avoid it. But there are things you can do to mitigate the risk. First, look for outlets and light switches - wires generally run straight up or straight down from them. The same principle applies to plumbing. Most electronic stud finders also have a feature to detect metal and live wires. They’re quite unreliable though.

Another precaution is to use masonry bits instead of wood or metal bits. Masonry bits have blunt ends, so you’re less likely to damage wires or pipes even if you hit one. You should also avoid drilling too deep, and whenever possible, aim for studs - there’s rarely anything critical inside them.

That said, hitting something important is quite unlikely. In a standard wall, there’s usually not much inside. Kitchens and bathrooms are a bit riskier. For perspective, I’ve probably drilled over a million holes into walls and ceilings, and I’ve only ever hit a wire once - and that was in the oddest place, running between a wall and a door jamb. It was such a strange location that the thought never even occurred to me until the room went dark.

If you hit a wire you’ll probably just blow a fuse. It’ll cost to fix it but it’s rarely catastrophical. Plumbing is a different story - you can cause a lot of damage in very short period of time. It’s generally recommended to locate AND TEST the main shut off valve before drilling if you’re unsure. Word of caution though: these valves may sit untouched for decades and may start leaking when you do.

The stud finder sucks, those give false positives for electrical all the time. You’re probably fine if the wall isn’t riddled with outlets, switches, or has a breaker panel.

If you’re really concerned, drill through only until you’re past the drywall. It’s only about a half to five eighths of an inch thick. Wrap a little piece of tape around your drill bit as a depth gauge. Alternatively, there are drywall anchors that look like fat plastic white screws that don’t require a pilot hole. You can get those and avoid drilling entirely.

If that detector is a recent purchase, might want to see if there’s any sort of manual included in the packaging or try looking up a brand name and model number. Most (if not all) that I’ve used have you start by holding the device to the wall to establish a baseline before doing a sweep across the area of interest. I usually try a few spots as the initial calibration point and also tap on the wall to listen for hollow/solid areas to see if the device readings are sane.

This stack exchange post has some useful tips as well. One specifically mentions touching the wall for grounding to reduce false positives, sounds like it could be related to what you’re seeing. There’s also a suggestion about adding layers of paper to decrease sensitivity so you’re filtering out the weak results that may be another source of false positives.

If in doubt, flip your breaker switch for that area and confirm with a plug-in tester before putting holes in stuff. It’s easier to turn that back on than to restart a human. As for anchoring, you might be fine with those little press-in plastic pieces but if your cat likes lasagna as much as some, might be worth checking out molly bolts or toggle bolts: they start with a bigger hole than the plastic bits but then expand on the other side of the drywall to spread out the applied forces. Over-engineering lets me down less often than just winging it.