Most recipes require that the roots be grated before use. Some recipes simply require it be sliced thin.
To freeze horse radish root for later use it should be grated beforehand and stored in an airtight container. Whole horseradish root will not always freeze equally and are more subject to freezer burn. Grated horse radish will not only freeze at a uniform rate but is much easier to defrost. If you have a vaccum sealer they work well for this purpose as they lock out the air and lock in the essential oils and aroma.
Enthusiasts will sometimes store / freeze whole roots in sand, although this is a bit burdensome for the average person it is effective.
Place the unmolested roots into a box of dry sand and place it in a cool, dark place throughout the winter. Fresh roots can be retrieved and used as needed throughout the winter season.
Fill a box with clean, dry sand, leaving about 1/4 of the total volume of the box as head space on top. Place the whole root into the sand as it would have been growing in the ground - the smaller side facing down. Do not allow any of the roots to come into contact with one another. Be sure the roots are entirely covered with sand and store them in a dark and cool / cold location. Don't allow them to come in direct sunlight.
Recipes for Horseradish Sauces and Vinegar
Basic Horseradish sauce
Creamy Horseradish sauce
Horseradish infused vinegar
Horseradish Greats: Delicious Horseradish Recipes, The Top 100 Horseradish Recipes
The leaves are always best when used fresh, they get soggy very quickly and do not store well. Fresh horseradish leaves are a spring treat, best harvested before summer sets in. Young leaves are subtle and tasty, older leaves of summer are tough and spindly.
Older leaves can still be used, but should be steamed or even boiled first along with spinach and /or assorted greens. The leaves can also be chopped and diced and added to some dishes in moderation. Pesto, dressings, soups, stews and so forth will benefit from a pinch of horseradish leaf.