which adds more value for a home, rental or otherwise?

Attached is usually preferred and is a sign of a newer house.   I think for rentals, it really doesn't matter.

From a rental owner perspective I'd prefer an attached garage. I know it could vary with location, I'd probably research the assessments in that area and see if there is a difference. I do believe if most of the properties on the block have attached garages and yours has a detached it could be worth less than an attached. But, where I live the assessor will give a higher value to the detached garage as a separate building..taxes are a bit higher for detached because it's considered a separate building but, insurance is a tad higher for the attached. Too many variables..either are better than none, that's for certain.