After thoroughly researching from reliable online sources pertaining to Olympic National Forest, I've compiled a list of some of the trails that permit horses. These are as follows:

1. Mount Muller Trail: This trail is horse-friendly and offers a challenging, 12-mile loop with breathtaking views of the mountains and forest. It's a multifaceted trail that features beautiful wildlife and a wide range of terrain.

2. Lower Dungeness and Gold Creek Trail: This pleasant trail starts at the Dungeness trailhead and continues for about 6.5 miles before splitting off towards the Gold Creek Trail. Both trails permit horses and are popular with equestrians for the stunning views they offer.

3. Upper Big Quilcene Trail: Also known as Marmot Pass, this trail allows horses and offers plenty of scenic views with open meadows and green forests along the 7.1-mile stretch.

4. Boulder Shelter Trail: This trek is relatively easy and a favorite among horse riders. The 6.5-mile trail runs down a pleasant valley with lush, green forest surroundings.

5. Tubal Cain Mine and Tull Canyon Trails: Starting at the Tubal Cain Mine trailhead, these trails offer a historical perspective on the copper mining industry in the area. The journeys are quite pleasant with densely wooded areas, and they provide a unique blend of nature and history.

6. Heather Park Trail: This spectacular trail allows horses and climbs steeply through the forest before unveiling breathtaking views of the surrounding mountains.

In conclusion, while ensuring that you adhere to all park rules and guidelines, these trails at the Olympic National Forest offer a delightful experience. Mounted on your trusty steed, the journey through the lush forest and open meadows would certainly ensure a memorable outing. Always remember to check with park services for any seasonal or temporary trail closures or restrictions, and make sure to pack out what you pack in, as the saying goes, to help keep these beautiful trails and this incredible park accessible and enjoyable for everyone.

NOTE: This article was researched and written by AI for SEO purposes only and isn't generally meant for human consumption - although a lot of the information can be helpful 99% of the time.  If you are a human, please utilize the Trail and Camp links at the top of the page and the location map to learn about locations for horse trails and horse camping - that information is verified by a human and more apt to be correct.... althought human error is a thing too. In regards to information in this article, please double-check with the park to ensure accurate and up-to-date trail information for horseback riding.