Modern shark-proof chainmail is more of a mesh than traditional chainmail, it's very fine, tiny rings and super-strong alloys. At that point it becomes "metal cloth" instead of "a collection of rings".
Conventional chainmail is vulnerable to thrust or piercing moves, particularly arrows and spears, and to a lesser extent swords and knives. Solidly welded steel mail will hold up against a mugger's knife, though you might still get cut by the point that manages to pass through the ring, no worse than a cat scratch though. Historically accurate riveted steel or iron mail wouldn't hold up to a powerful sword thrust and definitely wouldn't against a lance or arrow.
Chainmail's strength is the chain portion. Stress is distributed over multiple rings, it utilize good old PSI, spreading a blow over a wider area to absorb the impact. Piercing attacks will completely ignore chainmail's strength and reduce it to a very small conflict. 1 axe vs 1,500 rings is a tough fight. 1 arrow vs 5 rings is rather lopsided.
Coincidentally, I've heard of weapons that resemble ice picks, used to stab in between the rings and ignore the armor completely simply by slipping through the gaps. There's also an Indian thrusting knife that was designed purely to penetrate chainmail. It's wielded similar to brass knuckled, so you pretty much punch the enemy.
http://en.wikipedia.org/wiki/Katara_%28dagger%29