The effect is not that bad in this case, it is just a gray thin line, but I want to find a way to fix it, so the grid line extends from exactly 0 to exactly 10.

You can set PlotRangePadding to None in the X direction:

Plot[Sin[x], {x, 0, 10}, GridLines -> {None, {-1/2, 0, 1/2}}, PlotRangePadding -> {None, Automatic}]

If you are asking about the function for specifying the grid lines, some examples

Plot[Sin[x], {x, 0, 10}, GridLines -> {None, {-1, 1} &}]

Plot[Sin[x], {x, 0, 10}, GridLines -> {None, {-1, -0.75, 0.25, 1} &}]

Plot[Sin[x], {x, 0, 10}, GridLines -> {None, FindDivisions[{##}, 9] &}]

Thanks a lot Hans!

PlotRangePadding does solve the problem, but either way I do want to be familiar with this option to GridLines I mentioned.