Which of the several cover crops to use may be determined by what is available to you most easily and at the least cost, as well as the season you plan to grow them. At this time of year a cover crop needs to be winter hardy and that would include many of the clovers (annual clovers preferably) and Winter, Field, or Cereal Rye.
I use Winter Rye because I find that locally and relatively inexpensive at around $0.45 per pound. How much to spread depends on the area and the size of the seed, but Winter Rye is recommended to be applied at about 2 to 3 pounds per 1,000 square feet.
In the spring you can mow the top growth or as I do just walk over the plot and break the plant stems at ground level, which stops top growth, and leave those stems laying on the soil, as part of a mulch. The roots of many cover crops are much more extensive than the top growth and do provide as much, or more, organic matter to the soil as the top growth.

I tried winter rye once, many years ago in my veggie garden, and found that it was hard to remove in the springtime. I haven't used it since. Can you tell me more please about how you get rid of the topgrowth and root mass in order to plant?

I too like winter rye. My first spring with it I did find it nearly imposible to kill. I used a shovel and turned it over the full depth of the shovel. Not good. Since then I have found that if you till no deeper than 4 inches you cut the crown of the rye from its roots and it does die. You also leave the roots as nutrient holders and pathway for your new spring vegies. Mow it before you till or your tiller will 'wind' it up in a bail.
I also use and like annual rye grass mixed in with the winter rye at sowing. The two grow together without over competing each other. Looks gorgeous. The annual rye grass freezes to death over winter, leaves its roots and bulk, while the winter rye continues untill you kill it. I may try some radish mixed in next winter. Best regards.

I sowed oats on my beds for a cover crop and now it is about a foot high. Should I turn it under now so the soil microbes have food during the winter or just let it winter kill?
I use oats because it is inexpensive and geminates quickly. I usually let it winter kill and just rake it aside in the spring to plant, but when I look at all the green bio-mass now, I am wondering if I shouldn't dig it into the top inches of soil?

A cover crop should not be removed from the soil since it will add organic matter to that soil. As soon as I break the stems in the spring growth stops and the top growth is left as part of the mulch. One of the reasons for planting a cover crop is so your soil is not left bare, and exposed to the elements during the non growing season, so you would not want to turn it in now, wait until spring,

pagardener, thanks for your comments. This is exactly the info I needed.