Is there some condition on a complex algebraic surface that implies it has a smooth canonical divisor? I am searching for the sharpest possible condition, but sufficient criteria would be nice as well.


For example, the question is quite simple for surfaces that can be embedded in $P^3$. Roughly, just notice that in this case the linear system of the canonical line bundle is base point free and hence, by Bertini's theorem, we can find a smooth canonical divisor. 


When we cannot avoid having a singular canonical divisor, then we are left with some singular curves. I am interested in computing the Euler characteristic of their symmetric product. I will post a question about this in a follow up enquire. 



When is the canonical divisor of an algebraic surface smooth?
Any smooth projective surface with nonempty $K\_X$ is obtained by blowing up finitely many points on its unique minimal model. From the formula $K\_X=f^\*K\_Y+E$ for the blowup, you see that the exceptional divisors of the blowup are always in the base locus of $|K\_X|$. Thus, the problem is reduced to the minimal model.


Then you have to go through the classification of the minimal models of surfaces, that's been known for a hundred years now. (Using a book such as van de Ven "Complex surfaces", or Shafarevich et al, or Beauville...) 


For a minimal surface $Y$ Kodaira dimension 0 for example, $12K\_Y=0$. So either $K\_Y\ne 0$ and then $|K\_X|=\emptyset$, or $K\_Y=0$ and then any divisor in $|K\_X|$ is $\sum a\_i E\_i$, where $E\_i$ are the exceptional divisors of the blowups. 


For a minimal surface $Y$ of Kodaira dimension 2, the question is still somewhat tricky. If looking at higher multiples $|mK\_Y|$ suffices, then by a well known theorem (Bombierri? certainly I. Reider gave a very nice proof), $|5K\_Y|$ is free, so a general element is smooth (in characteristic 0). For $|K\_Y|$ I don't think the answer is known but why not search mathscinet.


Finally, for Kodaira dimension 1, an elliptic surface $\pi:Y\to C$, there is a well-known Kodaira's formula for the canonical class $K\_Y=\pi^\*K\_C + R$ with explicit rational coefficients in $R$. I'd play with that. Again, for higher multiples I think $|12K\_Y|$ works. 


Of course, to your example of a hypersurface in $\mathbb P^3$ you can add the case of complete intersections, and other surfaces for which $K\_X$ is either zero or $\pm K\_X$ is very ample.


