Trailing foliage is a great addition to any mixed hanging basket or container. They will add the interest of spilling out. Some of them even bloom :)
You can also use trailing foliage in garden beds, it will create a "carpet" along the soil or if you have a retaining wall it will trail out. There is a large assortment of colors as well as shapes of leaves

Dichrondria

Dichondra is a vigorous, fan-shaped silver foliage on silver stems; very heat and drought tolerant. Very strong trailing plant is a fun component in baskets and other planters. No pinching needed.
Full Sun to Full Shade

Silver Falls

Dorotheanthus 

 Dorotheanthus is a great plant to add to any garden or floral environment. Its high versatility makes it a wise choice as a border or edging plant. Create beautiful potted combinations, hanging baskets, window boxes, and garden landscapes with trailing red as a unique accent plant.
Full Sun to Full Shade

Mezoo Trailing red

Hedra 

Hendra is popular as an annual vine and houseplant. The lush, glossy foliage works well as a filler in annual combinations, or solo trailing from a hanging basket. This vine is very easy to care for as ivy can thrive in many different conditions, from full sun to partial shade. Consistently moist soil is considered ideal for ivy, but this plant won't complain much if you allow the top of the soil to dry out between watering's.
Full Sun to Full Shade

german ivy - green

German ivy - variegated

Ipomoea 

Ipomoea also known as sweet potato vines are a classic "spiller" plant perfect for container gardening. It has long tendrils that tumble over the edges and down the sides of containers. Although gorgeous in containers, sweet potato vines are versatile plants that are equally suitable for filling outdoor containers, spilling over a wall, or covering ground in a landscape bed. 
Full Sun to Full Shade​

Marguerite

solar power - black heart

SOLAR POWER - Red heart

Lysimachia

Lysimachia also known as Creeping Jenny is a fast-growing and vigorous groundcover, bringing mats of low-lying chartreuse color to gardens and containers. Its rounded golden leaves form on trailing stems with small, bright yellow flowers appearing in the summer. Perfect for softening hard edges, following a pathway or spilling from containers.
Full Sun to Full Shade

Goldilocks

Hillier sunburst

Ornamental Oregano

Herbs are one of the easiest plants to grow and provide pollinators a place to dine while livening up our dinners. Ornamental oregano plants bring all these attributes to the table as well as unique beauty and fun trailing form. The flavor isn’t as strong as the culinary variety but it has an unmatched appearance in its colorful bracts that develop in a host of pastel hues. 
Full Sun to Part Sun

Kirigami

Plectranthus

These fanciful foliage plants are easy to grow and make excellent houseplants or container plants. A great bedding plant, plectranthus can be used with flowering plants to add color with their foliage when other plants may not be in bloom. For best results, make sure to plant plectranthus in good, well-drained soil. One of the surest ways to kill a plectranthus is overly wet soil or pots that hold too much water. These plants are similar to a succulent and can handle short droughts from time to time. Even when the whole plant wilts, it will generally bounce back quickly when watered again.
Full Sun to Full Shade

Nico

Satureja

Satureja also known as Indian mint is a slightly unusual mint variety as it has a trailing habit whereas most mints are upright in habit. This evergreen perennial is highly fragrant and will produce blooms for many months. The flowers will be seen from early to late summer. Indian mint makes a good border plant and can also be used in baskets.
Full Sun to Full Shade

Indian Mint

Vinca Vine

The long, trailing stems of vinca major, provide a lush display of foliage. Lavender-blue flowers are an added bonus. Easy-care plant stylishly accents other bright colors.  Perfect for all kinds of containers, including window boxes, hanging baskets and combination planters. Tumbles beautifully over rocks, slopes, and banks.
Full Sun to Full Shade

Major Variegated