From How to Avoid Clothes Creasing During Wear:
Preparing Your Clothes
– Wait until your clothes have cooled to wear them
– Iron your clothes properly
– Ensure that you are wearing the proper size
– Choose your fabrics wisely (silk "cool" for many reasons, but it is worst for wrinkling)
– Purchase wrinkle-resistant fabrics (but not "plastic fibres" like polyester)
– Spray your clothes lightly with starch or fabric finish spray (controversial advice)
– Invest in a bottle of wrinkle-releasing spray (controversial because some of those chemicals and skin contact may not match)
Watching How You Sit
– Pull your clothes flat when you sit
– Avoid crossing your legs
– Don't put pressure on your clothes
– Stand as much as possible
That said, the higher the content of plastic fibres the less they tend to crease and wrinkle. But those fibres like spandex, nylon and polyester are also awful to wear on skin and especially so as underwear.
A compromise might be regenerated cellulose material like Tencel and Lyocell, that are similar to viscose but more robust, and made from plant material, with supposedly superior qualities to both cotton and polyester added on top. Since they use much less water, pesticides and energy during growth and manufacturing it looks currently as if these are indeed more eco-friendly also. People uninitiated to the quality of these fibres who like a soft touch to any fabric usually go crazy over these.
As of 2010, Lyocell is more expensive to produce than cotton […] main characteristics of lyocell fibres are that they are soft, absorbent, very strong when wet or dry, and resistant to wrinkles. (emphasis added)
Using wool for underwear might sound "nuts" to some. But wool is the best compromise in terms of wrinkle resistance, microclimate, and skin friendliness. The smaller the diameter of the fibre the smoother and less itchy these fabrics are. Wool from Merino sheep is where the good qualities start. Specialty animal wools might be an option for life hackers with more or too much money.
One thing to go for would be thicker material in general. Heavier cotton doesn't wrinkle as fast and heavier outer wear doesn't show wrinkles from underneath as much.
The more expensive solution for going cotton is to buy quality cotton. Egyptian, Pima, Supima or Sea-Island cotton would be marketing terms to look for. But what really matters is just staple length of the fibres used here. "Premium" or "Combed" approach the more expensive varieties. Cheaper cotton with short staple wrinkles faster and the wrinkles are longer lasting. Unfortunately it seems very difficult to find high quality cotton that is at the same time heavy. So I go for just heavy almost every time.