The most common trail used to Summit Mummy Mountain is the Lawn Lake Trail. To get to this trailhead, enter the National Park through the Fall River Entrance, Hwy 34. About 2 miles into the park, take a right where the signs point you to "Old Fall River Rd." and "End o' Valley." Once on this road, you will take your first right, where signs clearly point you to the "Lawn Lake Trailhead."
Follow the well established trail for 6.3 miles to Lawn Lake.
Just before reaching Lawn Lake, you will pass a junction with the Black Canyon Trail, which leads away to the right (east). Follow this well established trail for maybe 1/4 of a mile. Keep a very close eye out for a couple of small cairns on the left along a VERY faint trail that leads uphill to the north.
Follow the faint trail and a few more cainrs until the trail opens up and disappears. About halfway through this open area in the forest, look to your left for some more cairns, this time, with no trail. Follow these cairns steeply uphill through forest over loose terrain.
You will finally come to treeline and see the long, broad slope to the summit of Mummy Mountain. Although you can't see the summit from here, stay toward the left of the slope and just plug away uphill over many frustrating false summits. At times there are faint footpaths, but nothing that I would call a trail.
In my opinion, this route is better suited for descents. You can follow the slopes downhill fomr the summit, aiming for Potts Puddle, the beautiful emerald lake at the end of the slope just below treeline. Just before you would drop down the last of the ridge to the lake, a gully will lead downhill to your right, just as you reach treeline. It's here that you will pick up the cairns for the descent "trail" through the forest.
In the summer, just your typical hiking gear. A topo map will come in very handy while searching for the route through the forest.
Winter...snowshoes would be a must, obviously. If you were using this route for going up and down, you would not need any other winter tools.