It entirely depends on what you want to do with it and both have reasons why they exist.
The one where the top left corner is the origin and the positive xes point away from them are based on the way the pixels are stored, while the oke where y points at the sky uses a more mathematical approach. 2D systems thus generally use the former, 3d (or at least rasterization based ones) use the latter.
As mentioned by DMGregory, there's no handedness to these, because that system depends on the three axes. This is because a 2d plane divides the space into two parts and there's no way to tell by default, which should "contain" the normal vector.
Since rotation relies on trigonometric functions, it's also coordinate system dependent whether it's a CW or CCW system. When the angle is 0°, the vector you get is \$\vec v(cos(0°),sin(0°))=\vec v(1,0)\$, and at 90° it's \$\vec v(0,1)\$. In the "top-left origin" system this is clockwise, while it's counter-clockwise in the more traditional coordinate system.
One thing I must add is that even though I use the term "traditional" when describing the coordinate system where y points up, there's nothing stopping you, even in vanilla mathematics, from using other conventions. This is why you need to draw the arrows at the end of the axes.