Most water softeners have a bypass valve that will allow water to bypass the softener. Turn this valve, so water is not flowing through the softener. Return to the water meter and check the flow-indicator. If the flow-indicator is no longer moving, you have isolated the leak to your water softener.
You can also check for leaking swamp coolers, water-cooled air conditioners, ice machines, and reverse osmosis units by turning the bypass valve on each of these and then checking the flow-indicator at the water meter.