What ways do glaciers engage in mechanical weathering?

In short they carve their bed deeper into the country rock, using entrained rock to perform the grinding.

Glaciers are impressive 'rivers of ice' but on a photo of one you'll see continuous dark stripes along the length of a glacier.

These are moraines, rocks cast off by the mountains surrounding. When an avalanche occurs, it will sweep much other debris along with it as it makes its way to the valley /glacier floor.

Some of this rock material will work its way deeper into the glacier through crevasses and the like. At an ice-fall, where the glacier steps in its bed, crevasses will dominate the surface, and most of the moraine will become entrained.

The rock ground from the bed becomes as fine as flour - a rather muddy flour. Also, you'll be able to find striae, grooves carved in the bedrock by boulders grinding along on the base. Some striae may be seen in Central Park NY I believe!