Fix a prime p, and look at elliptic curves in some family (e.g. all elliptic curves ordered by height). How often do the Fourier coefficients a_p occur? Are there any conjectures?

There is a very clear discussion of some results in this direction in section 1 of Lenstra's paper "Factoring Integers with Elliptic Curves". I'll attempt to summarize. There are finitely many (about $2p$) isomorphism classes elliptic curves over $\mathbb{F}_p$. Most sampling methods choose the isomorphism class of $E$ with probability $1/\mathrm{Aut}(E)$. For example (in characteristic not $2$ or $3$), suppose you pick $a$ and $b$ in $\mathbb{F}_p$ at random and generate the curve $y^2=x^3+ax+b$, discarding it if it is singular. Then the number of $(a,b)$ for which you will generate a curve isomorphic to $E$ is $(p1)/\mathrm{Aut}(E)$. I imagine sampling by height will have the same effect. Weighting by automorphism groups, the number of $E$ for which $a_p=t$ is $H(t^24p)$, where $H(D)$ is the Kronecker class number. For $t = 2 \alpha \sqrt{p}$ with $\alpha$ fixed and $<1$, we are looking at $H(\Delta)$ for $\Delta = 4(1\alpha^2) p$. $H(\Delta)$ is $\Delta^{1/2+o(1)}$, but the oscillations are large enough to swamp the effect of $\alpha$. In some moral sense, one wants to say that we are converging on the distribution propositional to $\sqrt{1\alpha^2}$ as $p \to \infty$. In particular, it is true that the moments are approaching the moments of this semicircular distribution; see Birch. 


As others have mentioned, if $p$ is fixed then you're really looking at elliptic curves over a fixed finite field. From some points of view an interesting variant would be to look at elliptic curves say $E_{a,b}:y^2 = x^3 + ax + b$ where $a$ and $b$ vary over integers in a box, say $a \leq A$ and $b \leq B$ and relatively small compared to $p$. The one might try to find asymptotic results that hold as $p$, $A$, $B$ get large together. If $A$ and $B$ aren't too big then this is giving more information about individual curves. For example, in bounding the average analytic rank of elliptic curves it is important to get a good bound on $$\frac{1}{AB} \sum_{p < P} \sum_{a \leq A} \sum_{b \leq B} a_P(E_{a,b})$$ with $A$ and $B$ as small as possible. For example, see A. Brumer, The average rank of elliptic curves. I, Invent. Math. 109(3), 445–472 (1992). In a different but related direction, there is a paper of David and Pappalardi, Average Frobenius distributions of elliptic curves (it's the fourth from the bottom) on this subject. They get a kind of LangTrotter on average, so they are varying both $p$ and the coefficients defining the elliptic curves. Stephan Baier later made some improvements on this problem here. 


Wouter Castryck spoke about this at a GTEM workshop in Warwick. He considers isomorphism classes of elliptic curves over $\Bbb F_p$. His results are written up here: http://wis.kuleuven.be/algebra/hubrechts/DistributionTraces.pdf Of course there are a finite number of isomorphism classes of elliptic curves over a given finite field. So to answer your question for say elliptic curves over $\Bbb Q$ ordered by height, you would have to understand the distribution of the image of mod p reduction. I hope this at least gives a start. 

