Definitely a stitch picker is the way to go. Sometimes, the stitching is very tight and small and the labels are usually doubled over at the ends. Cut off the main part of the label (the section between the stitched ends), as close as possible to the lines of stitching. You will be left with the two rows of stitching and doubled over remnants of label. With the picker, work at the label fabric's cut edges to make them fray. It should be possible to gently pull off/cut off frayed strands until eventually, you have little or no label remaining, just the stitches. Then cut the stitches and tease away the thread.
To minimise the danger of making holes in the garment, resist the temptation to pull the label taught to get at the stitches.