It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only ordinary singularities, i.e. a curve $C$ of double points, containing a finite number of pinch points and a finite number of triple points, which are triple also for $S'$. The proof is obtained by embedding $S$ in $\mathbb{P}^5$ and by taking a projection $$\pi_{L} \colon S \to \mathbb{P}^3,$$
where $L \subset \mathbb{P}^5$ is a general line. This is the method originally used by M. Noether in order to prove his famous formula $$\chi(\mathcal{O}_S)=\frac{1}{12}(K_S^2+c_2(S)),$$
see Griffiths-Harris Principles of Algebraic Geometry, p. 600.
Question. Is it also true that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only isolated singularities? And if not, what is a counterexample?
Best Answer
IMPORTANT EDIT 12-2015
There is this paper of Tokunaga "Irreducible Plane Curves of Albanese Dimension Two" which based on cited work of Kulikov constructs surfaces in P^2 with isolated singularities and Albanese dimension two. In other words, the conjecture that for a smooth model of a normal in P^2 the Albanese dimension should be one is false. This makes the answer to the question of the OP open. This information was given to me by R. Gurjar.
Original Answer- To the best of my knowledge this is a long standing open problem. I cannot recall a reference, as this is something I studied in the 1980's, but I recall this being phrased as an unsolved problem from the 19th century Italian school. The conjecture is that no normal surface in P^3 is birational to a smooth surface which has two dimensional image in it's Albanese. One specific case of this that has been studied more extensively are Zariski surfaces:z^n = f(x,y) where f is a polynomial of degree n with only cusps and nodes as singularities. There are lots of information about when such a surface is irregular, but beyond that not much is known. I believe that even if f is a sextic polynomial it is unknow whether or not the resulting surface can have 2 dimensional image in it's Albanese. I have heard Catanese ask about the case where S is an abelian surface.