The Ödkarspitzen are located in the Hinterautal-Vomper chain of the Karwendel. It's the second highest peak of the Karwendel, though merely 4 meters smaller than it's direct neighbour to the east, the Birkkarspitze, highest peak in the Karwendel.
The Ödkarspitzen consist of three peaks, the Westliche Ödkarspitze (2.712m, western peak), the Mittlere Ödkarspitze (2.745m, central peak) and the Östliche Ödkarspitze (2.738m, eastern peak), only seperated by minor gaps and ranged by an exposed but partly secured trail (UIAA difficulty II).
The ridge trail and therewith all three summits can be reached by three different ascents, described further below. Sure-footedness and freedom from giddiness are absolutely essential for any of the ascents to the Ödkarspitzen.

North face of the Ödkarspitzen.

Getting There

From Scharnitz in the western Karwendel you can choose eighter the Karwendeltal to the north-east and climb the Ödkarspitzen from the north, or the Hinterautal to the east and climb them from the south. The Karwendeltal is definetly the better choice, as the ascent from the valley is hardly done at a single day and the Karwendelhaus offers a perfect stopover, roughly half the way up (5:00 hours).
From Hinterriß in the northern Karwendel, close to the german border, you can ascent via the Johannestal to the south, which is also leading to the Karwendelhaus (4:00 hours).

Routes to the Ödkarspitzen.

Red Tape

No permits, no fees, no closure.

When To Climb

July to September the ascents should be free of snow. Otherwise light crampons can be fairly usefull. Unexperienced climbers should take along a harness and via ferrata kit for self belay at several steel cables.

Camping

The Karwendel is a nature reserve, so camping is strictly prohibited.
You can stay at the Karwendelhaus during summer and autumn. Meanwhile they also got a winter room, which is accesible for everyone during winter time. In case of need a bivouac (Birkkar-Not-Biwak) is located at the ridge between the Birkkarspitze and the Ödkarspitzen.

Routes

1) Schlauchkar
3:00 hours, 974 m gain in altitude
From the Karwendelhaus (1.771m) ascent to the south along the Schlauchkar (gravel and debris). This route is getting really steep at about 2500 m, leading across a notorious snow field (50°) which lasts long into the summer time. Anyhow it's the easiest access to the peak.
From the gap between the Ödkarspitzen and the Birkkarspitzen there's a trail along the ridge to the west. There are two or three very steep and exposed parts, though secured (UIAA difficulty II+).

2) Westliches Birkkar
approx. 5:00 hours, 1.545m gain in altitude
From the Hinterautal (1.200m) in the south a fairly long trail is leading to the gap between the Ödkarspitzen and the Birkkarspitze. From thereon, climb the ridge trail described above.

3) Brendelsteig (for experienced climbers only)
5:00 hours, 974 m gain in altitude
From the Karwendelhaus (1.771m) ascent through the Schlauchkar, but choose the turning "Brendelsteig" to the west at about 1.900m. Traverse the valley, climb up to the top of the Ödkarspitzen north ridge and follow this just below the wall at 2.300m. Traverse to the Marxenkar on the right (west) and zigzag this terrain to the Westliche Ödkarspitze. Follow the ridge trail to the other to central and eastern peak.
This trail is very exposed! Some parts are secured with steel cables. Sure-footedness and freedom from giddiness are absolutely essential!
Don't climb it in bad weather. There had been some major accidents while thunderstorms, claiming several lives.

External Links

Steinmandl.de
Route description (in german)

Karwendelhaus
Website of the Karwendelhaus-Hut