It grows well in most climates although in warmer climates it may need more water. It spreads by sending out rhizomes, or little creeping vines. This is beneficial because the hardness of red clay makes it difficult to grow from seed.
Bermuda is another creeping grass. It is well suited to hot and warm weather climates. It is very drought resistant and will thrive and grow rapidly in red clay.
Zoysia is a creeping grass similar to Bermuda. It will grow well in warm areas but is not nearly as drought resistant as Bermuda. It will hold up to heavy traffic well and can be considered if you have children or pets.
St. Augustine grass grows on a vine-like apparatus across the lawn. It is not the best for heavy traffic and can survive in soils with less nutrients. The advantage it has is that it will hold up well in coastal regions where the temperature can drop at night. It will need weekly watering but will thrive in red clay.
Centipede grass is another grass that will grow in red clay. However it should be noted that it is a slow grower and also is not tolerant to cold weather. It never goes truly dormant and is therefore exposed to the environment. It is primarily a warm weather grass and will need regular supplemental watering. It is not as salt tolerant as Bermuda, so it should not be used in areas too close to the ocean.