Most miniatures are cast with a two-part mold. Even the best casting processes usually leaves a line behind where the two mold halves meet and a tiny amount of casting material seeps out. You want to remove this before priming or painting your miniatures.
Any remaining flash tends to show itself even more once you prime your miniature, so you want to remove it before beginning to paint.
If the mold lines are not thick, you can remove the flash by dragging a sharp X-Acto knife lightly along the mold line. Use a motion which draws the blade away from your fingers, hands, and body to avoid injury.
For all but the most minor mold lines on metal miniatures, I use a jeweler's file. These thin "needle files" have fine abrasive, often coated with diamond grit. These files help avoid damaging the miniature when removing mold lines. You can find them at most hobby shops or hardware stores. You can find these jeweler's files both straight and curved. I own both kinds, but when getting into hard-to-reach or curved areas, I prefer the curved type.
If you're serious about painting great miniatures, you owe it to yourself to spend the $10-$15 for a good set of needle, or "jeweler's files." They allow you to remove mold lines in small areas without damaging surrounding details.
These files come in many shapes, including round, square, rasp-shaped, and square. I find the round and rasp-shaped (one edge is rounded, while the other is flat) the most useful.
Take care to use a file which most closely matches the area where you find mold lines. For flat areas, such as bare arms or legs, I use a flat file. If I need to get into small crevices, narrow files help me remove the line without sanding off fine detail. With a little practice, you'll see what I mean.
Plastic or resin miniatures are cast with materials which are much more delicate and easily damaged than metal miniatures. For plastic miniatures, you can typically remove all mold lines with a sharp Exacto knife used as described above.
However, with these softer casting materials, you must use a light touch or you risk cutting notches or removing detail from an otherwise beautiful cast figure.
In order for casting material to flow into a mold without leaving air pockets, molds often have small openings cut into them to let air and some casting material flow away from the figure itself.
This leaves little spurs, often found on feet, hands, or other areas where casting material might not otherwise flow. Remove these with an Exacto knife with a motion away from your hands or body. Clean them up with a jeweler's file.
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To make the lines and details of a mini as sharp as possible, one trick that I have found works extremely well is to lightly drag a sharp dental pick along these lines.
When you apply paint, it's difficult to avoid at least some details being lost, since paint tends to want to settle in lines or where surfaces meet.
When miniatures are cast, especially if the mold has been used many times, sometimes detail gets lost. Just a few strokes with a dental pick can help make fine details stand out, especially if any release agent remains after cleaning and preparing a miniature for painting.