It's up to yourself really. Having a solid background might restrict it's usage in some ways. For example, you could have a 15% tint of the foreground colour as the background. In print circles it is just one colour. In embroidery it is 2 colours.
Most people tend to cater to both sides of the scenario. A full colour, and single colour logo, how both appear on a transparent background, how both appear on a solid background.
Look up logo brand books. There are lots of things to consider. Like minimum size, use of a monogram or emblem if too small. Colour do and do nots. Minimum clearance around the logo. Preferred placement. How it's placed with other logos. etc