Which is safer for monitor: Turn off or Screensaver?

Why kind of monitor? Who makes it? What is the model?

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If you are talking about physically turning off the monitor with the Power Button, then absolutely the screen saver is the only way to go.
Turning electronic components on and off all the time is stressful to them.

If you like screen-savers for their aesthetic appeal, then by all means, use them; however, keep in mind that the CPU is still doing a lot of work to display all those fancy graphics, so it is not a replacement for the long term- put your computer to sleep instead.

There has been decades-long discussion about turning off a computer when done with it, or leaving it running all the time.

This is just my opinion, but I have come up with a happy-medium of about 4 hours.
In other words, I think if you are not going to be using your computer for 4 or more hours, then turn it off.
If you are going to be back at your computer before that amount of time, leave it on (or put it to sleep).
If you use a CRT, then use a "Blank" screen-saver to help prevent burn-in.
I've never understood the reasoning behind having a graphic-intensive screen-saver running if nobody is there to watch.
Seems kind of a waste to me.

Ultimately, it is each individual's decision as to what constitutes a 'reasonable amount of time'.

Just some thoughts, folks...

WinCom7 is right!
It's up to each individual and the time they will be away from the monitor.
I usually turn off the monitor if I have to leave and I am in the midsts of backing up my pc.
That way I can multi-task like backing my system up while mowing the lawn or other "Honey Do's!"
But what works for me may not work for others.

Not really a question of safety but aesthetic and convenience. As mentioned above the screen saver protects the image burn of early CRT's and early flat panel displays. Today I don't believe this is such a concern with led and lcd technologies but we are still used to screen savers and they are nice and customizable. Screen sleep modes are also an option as they turn the display charge off but still await an instant on feature so it uses less energy but it's still on and consuming some. The best energy use would be to turn your display off when you walk away and then power back on when ready. This is what I do, so screen save when getting a second cup of coffee and turn off when heading out or away but due back in a reasonable period of time.

i have never had a monitor fail on me and i always have windows turn off the display after 5 mins, i would say turn the monitor off as its a waste of electricity and as others said whats the point of a screensaver when no one is there to see it?

In my case, as others have stated, it depends on how long I'm going to be away from the computer. Several hours and I put the computer to sleep and physically turn off the monitor. But then, I'm paranoid about some things and have read about house fires from computer monitors. I've never met anyone personally that's had this experience, but I'd hate to lose my dog and 2 cats.

Belkin has come out with some cool little power saving widgets that look to be timely as our power rates continue to climb.
Belkin : Conserve