The Idea behind cover crops is to keep living roots in the ground. Ideally, these are the types of roots that benefit the soil for fruits & vegetables. Good cover crops provide sustenance for soil microbes. Sure you can jump start things by feeding bacteria with compost but you must keep feeding then whereas cover crops do it for you by providing the soil new nutrients from the sun daily.
Perennials are the best cover crops because they create long term relationships with fungus which in turn build and support those underground urban communities of bacteria. It's all very complex which is why you want living things to do the work for you. Do you want to be constantly testing your soil for PH, nutrient balance, moisture etc.?
In my vegetable garden I want my cover crops to fit few categories: I want them to naturally grow shorter than 3-4”. All plants will grow huge when provided ideal conditions but if you can carpet your garden with low growing perennials, they'll keep the soil alive until you plant something that quickly overshadows any competition. These can be field herbs (weeds), flowers, or vegetables. In my book, the definition of a weed is a non edible invasive plant whos roots add very little to the soil yet inhibits the health and vitality of other plants by robbing nutrients and crowding them out. Bindweed and Pig weed are the only two “true weeds” I know. Everything else is a “field herb” either for it's edible or medicinal value or because it actually amends soil. Hoary cress and horse weed are edible but not tasty
Cover crops can also be short season vegetables like radish or can even be peas or beans used at the end of a season and used as succession crops only for cover and no crop. In my food garden I also like cover crops to be edible, because why not? There are lots of options, why not start with the edible ones?
Of course, there are super robust exotics such as Japanese pachysandra (Pachysandra terminalis) and Vinca minor (commonly known as periwinkle) that will quickly spread to cover the soil and are almost carefree. Unfortunately, their spread doesn’t stop at the property line — they commonly invade adjacent natural areas, displacing the native flora. Fortunately, the native flora routinely offers many robust alternatives.
Unlike traditional groundcovers, which are commonly planted in single-species stands, a native green mulch should be diverse. In other words, don’t just plant black-eyed Susans (Rudbeckia hirta) in that sunny soil around the base of your shrubs or trees, intermingle it with other plants that are commonly found with it in the wild, such as bee balm (Monarda fistulosa), purple prairie clover (Dalea purpurea), and blazing star (Liatris spicata). In this way, you will greatly increase the types of wildlife your planting will benefit, and by providing flowers that bloom at different seasons, you provide an extended buffet for pollinators, as well as prolonging the treat for your eyes.