From Hadamard's bound the largest possible determinant of an $n\times n$ (0,1) matrix is $h_n=2^{-n}(n+1)^{(n+1)/2}$.   The data at http://www.indiana.edu/~maxdet/spectrum.html suggest several conjectures:

 1. The spectrum is "dense" up to a certain point, after which it becomes "sparse".  An integer in the "dense" part is almost certain to be the determinant of some $n\times n$ (0,1) matrix; and integer in the "sparse" part is almost certain not to be.  The point at which the spectrum becomes sparse is asymptotically some constant times $h_n$.  The data suggest that the constant is near 0.5.  I think this is basically the conjecture made by Gerhard Paseman in his answer.
 2. A stronger statement is as follows: let $g_n$ be the position of the first "gap", that is, the first positive integer that is not the determinant of some $n\times n$ (0,1) matrix.  Then asymptotically $g_n$ is some constant times $h_n$, and again the constant appears to be close to 0.5.  If $D_n$ denotes the set of positive $n\times n$ determinants, and if the above ideas are on the right track, then it seems likely that asymptotically $g_n/|D_n|$ is 1.

I don't have even a heuristic explanation as to why such conjectures ought to be true, and one can always worry about how much one should try to conclude from data that, in fact, go only as high as $n=21$.  As mentioned in some of the comments to other answers, $D_n$ is really only known for $n\le 10$ and $n=12$.  The sets $D_n$ for these $n$ are given at the above link, as are conjectures for all $n$ up to 16.  Data for $17\le n\le21$ have not yet been added to the site.  (Note that the $n$ on the web site refers to $(-1,1)$ matrices, so one should subtract 1 from it if one is talking about $(0,1)$ matrices.)  I do have high confidence that the listed sets $D_n$ up to $n=16$ are not missing any values, even the ones that have not been proved complete.