With a miscarriage, the woman may choose to go naturally, use herbs or medication, or have a D&C. Depending on how far along she is, she may or may not pass tissue that is recognizable as a baby. If she has a D&C, she will likely never see anything (unless she has the D&C for retained products of conception, and has already passed the baby). In many states and many hospitals, the woman is not told what happens to the baby, nor given an option for receiving the body and taking it home to bury it, if the fetal demise is prior to 20 weeks (after 20 weeks, it’s considered a stillbirth). One woman worked to change the law in her state when she was told — too late — that her 16-week baby went into the medical waste disposal. (There are multiple links in the comments, so scroll down to read the full story, with more information.) The baby would be about 4-5″ long at that gestational age.
When a woman has an early miscarriage, it may be difficult for her to catch the remains, even if she wants to, unless she sits on a bucket all day, or puts a colander inside the toilet. In my case, I felt the biggest clump of tissue pass just a few minutes after using the bathroom, so it very easily could have fallen in the toilet. As it was, it was easy for me to have it, examine it, and eventually bury it. Not everyone is comfortable with that, and don’t think anything of just flushing the toilet. If the woman experiences labor-like cramps or contractions, it will be easier to catch the remains than if she just has period-like bleeding with no cramps, because there is a warning that it is imminent. Several of the women I know who have had miscarriages (particularly the later losses) have buried the remains; one woman cremated the remains; and many others did not keep the remains at all.
But with a D&C, that is often not an option at all. If you have a D&C, you can ask the doctor if you can have the remains. It’s possible there might be some sort of law against it, or hospital procedures making it difficult to do [I remember some case out in Hawaii in which a couple had to sue the hospital and win before they would turn over the placenta to them after a birth, so anything’s possible!], but you can at least ask prior to the procedure, if you wish.