Is it safe to leave butter at room temperature?

Try a butter crock. This will keep your butter fresh at room temperature for a fairly long time (weeks, not days).

It depends on the room temperature where you live. At 65F (18C) or below, butter is often barely spreadable and will last for weeks on the counter in a sealed container. At 80F (26C), it starts to get overly soft and doesn't last more than several days.
Our family goes through about a pound / week and we've never had any issues with keeping a half-pound block on the counter at any given time - we finish it off before it has time to lose any quality or flavour. The rest we keep in the fridge until needed.
The most important thing is to keep it in a covered container - I'm sure a butter crock would do a great job, but even just any old small glass container with a lid will do.

The question seems to have been more about food safety than whether it seems palatable.

When the fat in butter decomposes (i.e. when the butter becomes rancid), it produces an unhealthy acid that actually inhibits mold growth. So, don't wait for your butter to mold to determine if it's gone bad.

To follow strict food-safety guidelines, protect butter from heat, light, and air; store it up to two weeks in a refrigerator, below 40 degrees.

It can also be frozen for 6 to 9 months.

As long as you use salted butter it will keep in a covered container at room temperature for at least 2-3 weeks without getting mouldy or rancid, in my experience. If you use unsalted butter there are more microorganisms that can live on it so it spoils faster, but there aren't any common contaminants that can grow on salted butter other than moulds, and even they grow very slowly on it.

I keep my butter in a covered dish next to the toaster. When it gets hard on the outside I toss it. This doesn't happen very often as I am now using 1/8 pound sticks. Usually the sticks last about a month in the summer, longer in the winter. If it has been a while I'll smell it before using it or just toss it.

When I first got a microwave oven I tried to use it for warming the butter when I took it from the fridge. I found that the butter went rancid if you did it two or three times.

Although just microwaving a small portion to use was OK, it was difficult to time the warming so the stuff didn't melt. Now I just keep it in a butter dish at room temperature, except in high summer.

I have always kept my butter, 1/4 pound at a time, out on the counter in a covered, pottery-type butter dish (Fiestaware) or a covered glass dish. The latter is probably less desirable because of light exposure, but either way, I have never had a problem, and I am picky about food freshness. We use the 1/4 pound within about a week, I'd say.

The exception is in summer, when it sometimes gets hot enough to melt the butter in the dish. At those times, I put the butter dish in the wine refrigerator, which we keep at 55 degrees F. If you happen to have a wine refrigerator, it's a great compromise - the butter doesn't spread as easily, but it isn't rock-hard either, and it's better than having it melted.

Yes. Butter is cultured cream, meaning there are good bacteria fighting off the bad bacteria. Cover it in a butter crock or similar and it should last a couple weeks.

I have left butter out on the counter uncovered for as long two or three weeks or more. It has never gone bad are tasted any different . We do eat butter everyday so we use it quickly. Sometimes I microwave it when I first take it out of the Frig. if I'm going to use it right away. I come from a family that has always left the butter out and we never noticed a difference in the taste or had butter go rancid. The only time I had mold on butter is when there was a hurricane and the electricity was out for weeks and we had to throw out every thing in the Frig. I use salted butter but occasionally unsalted with no problem.

My experience is that butter crocks just suck. I was all excited when I first had one (what a great idea!), only to find that the butter would get really yucky tasting and even moldy quite quickly.

Then maybe fifteen years later (last year) I decided to try again. Same experience.

I've also tried it without the water -- similar results.

A simple covered butter dish is much more effective, in my experience. Butter keeps and tastes good for quite a while -- a couple of weeks, anyway.

I have no scientific theories to explain this (though I'd be vaguely interested to hear some). Just the empirical facts, validated through repeated experiments with consistent results.

Room temperature is the best for maintaining the shelf life of food materials. We also have to maintain humidity. It should not be more humid because it can produce a favorable environment for bacterial growth. Lower the temperature more will be the safest.