Use the Beaver Brook Trailhead at Kinsman Notch described in the Getting There Section and make sure you know about the Parking Pass described in the Red Tape section.
The 3.8 mile class 2 Beaver Brook Route (part of the AT) is one of the shortest and possibly most scenic route to the summit. The trailhead is just off of NH SR-112 at Kinsman Notch (1870') just west of Lost River. Take the Beaver Brook Trail west from the road quickly passing the scenic Beaver Brook Cascades where the trail has been improved with stone steps, wood steps, and hand rungs. The cascades end 1.1 miles from the trailhead and in another 0.4 miles you'll reach the DOC Beaver Brook Shelter (3750'). Reach the Benton Trail in another 1.9 miles during which time you'll pass the Asquam Ridge Trail junction and Mount Blue on your right (north at this point). From here take the Benton Trail 0.4 miles up Moosilauke's exposed north ridge to the summit. For reference, use the trail map under the Getting There section.
It's worth noting that Topozone still indicates the old route of this trail. Just after crossing eastern lip of Jobildunk Ravine, there's a small sign where the new route makes a sharp rightward zigzag and heads north to a point just west of the summit of Mount Blue, before heading west to the Benton Trail. The old route, as you can see on the map, and which is still visible on the ground, continues to traverse the edge of the ravine and eventually joins the Carriage Road to reach the summit from the South. (Ref: nartreb)
None during the snow-free season, however gear such as ice axe, crampons, snowshoes, and skis may be used, or even mandatory, during winter conditions.
If you have information about this route that doesn't pertain to any of the other sections, please add it here.