Well you're going to get answers all over the place for this one, it is a much loved topic to kick around. The short answer is there are many answers, depending on many factors. Chiefly, that is how you use your boat, your local conditions and your preferences. No one answer is right for everyone, you have to decide what is right for you.

And you have to be very specific. 9/16" nylon what? Double braid? Three strand? Eight plait? All have different strengths and characteristics.

What is right for us is 150' of 5/8" double braid with a steel thimble shackled to 100' of 5/16 G42 hi-tensile chain. I use 3/8" Crosby shackles that are rated as strong as the chain. The shackles are all safety wired. We have fairly deep anchorages, none have total coverage and we get a lot of wind, frequently from bad directions. So we also have a 45 lb Manson supreme anchor which is 1 size up from what you could use in quieter conditions.

Personally, I think 1/4"HT is too small for these boats and 5/16"BBB not strong enough, 5/16 G42 is the best combination of weight/strength. Carrying a lot of weight way up forward is a very good way to kill the boat speed on a 36, speaking from what I've learned here from our racer folk and personal experience. This works for us, others have different requirements and this wouldn't work for them.

Depends on your model of windlass also.
Mine is a Maxwell Freedom 800 and it will accommodate both chain and rode without removing from the capstan. Therefore it specifies certain dimensions. I use 5/16" G40 chain and 5/8" three strand with an anchor-to-rope splice...