A hard, unripe plum lacks sweetness and tastes astringent. Choose plums that yield slightly to a press of the fingers. The softest point should be at the place opposite the stem end. Plums do not ripen once removed from the tree. You may be able to soften them slightly by placing them in a paper bag with another fruit such as a banana or apple, but you won't appreciably change their sweetness.
Other Ripeness Indicators
Choose fruits that are heavy, which indicates they have a lot of juice and dense pulp. Fruits that feel very soft usually are too ripe. If you're picking them off a tree, they should release from the stem with a gentle twist. Plums that are hard to pull from the tree are not yet ripe. Store ripe plums for up to two weeks in a refrigerator set at about 32 degrees Fahrenheit.