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Sheet. White is the most obvious shade to use, being completely neutral. However, especially if you're doing portrait photography, you may want to try a slightly off-color shade of white for a warmer feel.
If you do not have any white sheets, and/or don't want to buy one expressly for the purposes of a photography prop, check out second-hand shops. A few stains or holes here and there certainly won't ruin the overall effect, though it might not make you appear particularly professional.
Natural Lighting. There's nothing quite like a sunny day!
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Here's the tricky bit.
A sheet obviously can't support itself, so you need to come up with a means of supporting it. Preferably, this can just be a friend of yours with a bit of height on their side, which is both easily adjustable with a little communication, and mobile, as your friend will likely have legs that they can walk around with. Otherwise, you might need to get creative with strings and existent supports or tripods.
The beauty of using a sheet is that it doubles both as a reflector and a diffuser, as well as being a handy background. Vary the positioning of the subject relative to the sheet to use it in its different capacities.
As a reflector, the sheet should be situated such that it will bounce the light back at the model at the desired angle. The model could sit or otherwise pose upon it, for instance, so that the light bounces up at their face and eliminates unsightly wrinkles. Or, if you position the model in the shade, the light could be reflected from a patch of sun onto the model's face and thus bring in some much needed dynamism.
As a diffuser, the sheet needs to be placed directly between the model and the light. This will soften the harsh contrast between sunlight and shadow that occurs on a sunny day, and will create a generally more pleasing picture. Different weaves will diffuse the light differently, so you may want to try a few different sheets out to get the desired effect.
As a background, the sheet will provide a wonderful glow to the image, as well as providing a simple, minimalist background with beautiful natural lighting. Be careful to also reflect/diffuse light into the model's face if you do this, creating a close semicircle around the model, as otherwise it may appear too dark relative to the background.
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Variations & Resources
Of course, you can get a bit creative with the sheet. It doesn't need to be a neutral tone. Try using sumptuous reds or deep greens that harmonize or contrast with the subject of the portrait. Or, if you want to impart a particular mood upon the shoot, you can use printed sheets in the background with designs like faded paisleys to impart a sense of nostalgia, or bright polka dots for a bit of fun.
There are other materials you can experiment with, of course, in you're in an innovative mood and don't feel like this sheet quite cuts it. More properly reflective materials can provide a less diffuse lighting effect, such as aluminum foil, though this will require some innovation with regards to framing to make a proper DIY reflector. Sun shields from car windshield work well, for instance. Consider using naturally stiff materials like foam board, particularly white foamboard. Some photographers have even bounced light off of their own white t-shirts as a reflector! There are many possibilities to explore.
For an excellent brainstorm on some techniques associated with using a sheet as a reflector, check out this tutorial from Digital Photography School. The comments afterwards also provide many interesting ideas.