Are most rational quintics unsolvable?
Just to get a data point, using Maple I took $2000$ random quintics with coefficients pseudo-random numbers from -100 to 100 (but the coefficient of $x^5$ nonzero). $1981$ of these were irreducible (of course the reducible ones are solvable). All $1981$ irreducible quintics were not solvable.
EDIT: Quintics with a rational root are solvable, and these are easily seen to be dense in $\mathbb Q^5$. Namely, take a rational approximation $r$ of a real root of the polynomial. Then $p(X) - p(r)$ has rational root $r$, and is arbitrarily close to $p(X)$.
EDIT: If I'm not mistaken, quintics with Galois group $S_5$ are dense in $\mathbb Q^5$. Consider the proof that $x^5 - x - 1$ has Galois group $S_5$. The same proof should apply to $p(X) = X^5 +\sum_{i=0}^4 \alpha_i X^i$ as long as
- All of the denominators of the $\alpha_i$ are congruent to $1 \mod 6$.
- The numerators of $\alpha_0$ and $\alpha_1$ are congruent to $5 \mod 6$, those of $\alpha_2, \alpha_3$ and $\alpha_4$ are congruent to $0 \mod 6$.
$5$-tuples satisfying these conditions are dense in $\mathbb Q^5$.
Yes, in fact, we can generalize Mike's reformulation (with integer coefficients, and allowing nonmonic polynomials, which ought to be inessential) and give a stronger result: Let $P_N$ denote the set of monic polynomials of degree $n > 0$ in $\mathbb{Z}[x]$ whose coefficients all have absolute value $< N$. S. D. Cohen gave in The distribution of Galois groups of integral polynomials (Illinois J. of Math., 23 (1979), pp. 135-152) asymptotic bounds for the ratio in the above limit. Reformulating his statement with some trivial algebra gives (at least asymptotically) that $$\frac{\#\{p \in P_N : \text{Gal}(p) \not\cong S_n\}}{N^n} \ll \frac{\log N}{\sqrt{N}},$$ and the limit of the ratio on the right-hand side as $N \to \infty$ is $0$. This implies a fortiori for $n = 5$ that $$\lim_{N \to \infty} \frac{\#\{p \in P_N : \text{Gal}(p) \text{ is solvable}\}}{N^n} = 0,$$ since for quintic polynomials $p$, $\text{Gal}(p)$ is unsolvable iff $\text{Gal}(p) \cong A_5$ or $\text{Gal}(p) \cong S_5$.
Some similar results were produced a few decades earlier: B. L. van der Waerden showed in Die Seltenheit der Gleichungen mit Affekt, (Mathematische Annalen 109:1 (1934), pp. 13–16) that the above ratio has limit zero (at least when one allows nonmonic polynomials and adjusts the denominator accordingly, which is probably inessential).
For more see this mathoverflow.net question and this old sci.math question.