Both of my machines under my desk have issues with their graphics cards. One pastes rows of coloured dots over part of the screen, not always the same place, and they can show variable colours, not trusting the memory, I do not allow GPU work on that system. The other has a noisy fan, which doesn't really bother me, but drives my wife crazy at night, so I switch it off until the morning. They are ASUS HD 7850 cards. So am thinking of replacing both and am here to ask for things to look out for when considering the new examples. I don't play games etc. so that is not an issue, I want cards that are quiet, long lasting and good BOINC crunchers. I open the floor...

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Sorry, just thought I'd add, the screen plug has a rectangular plug like a din plug with three rows of pins .
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Wave upon wave of demented avengers march cheerfully out of obscurity into the dream.

the largest factor will be your budget. how much are you willing to spend?

does your monitor accept different inputs? what you described is the VGA connector, which is the old analog. no modern GPU supplies this anymore. does your monitor have DVI? or HDMI? HDMI would be ideal since most modern GPUs still have this connector, but even DVI would be OK since you can adapt display port or HDMI from the GPU to DVI at the monitor quite easily.

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The monitor has a smaller socket on the back next to the one I am using. It is about a third of the size of a din socket but has a similar shape. It is a bit difficult to see now as it is dark, and the screen shades the rear from the room light.

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Wave upon wave of demented avengers march cheerfully out of obscurity into the dream.

Check different types from here: https://en.wikipedia.org/wiki/List_of_video_connectors

I think the other socket is an "hdmi" type. There appear to be 3 types of hdmi, I am not sure which it is yet, I'll be able to see more in the morning. The cable I am using now looks like "dvi-d" on that list.

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Wave upon wave of demented avengers march cheerfully out of obscurity into the dream.

You should be good to go once you purchase a HDMI or DP to DVI-D cable or adapter.

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