I use an Xacto Knife for veneer and fine cutting. The tip of the replacement blade wears out all too quickly
This is the problem drawn for clarity
The top two drawings show a side and top view of a new blade. The middle two show a worn blade (exaggerated). The fine point is now a larger flat (or more accurately rounded) point and will also be burred over slightly (not shown) making it even duller. What I do to restore the point is grind the back of the blade off, at a slight angle, on an oilstone, to past the blunt material (bottom two drawings). This can be done several times and whilst the attacking angle of the tip of the blade is a little broader at least you have a freshly sharp tip to work with.
Be seeing you.
-- Martyn -- Boxologist, Pattern Juggler and Candyman of the visually challenging. http://www.theartofboxes.com