(Per request, post edited to discuss $G_{n,p}$ for values other than $p=1/2.$ This is a routine line of reasoning in probabilisitic combinatorics. See the first few chapters of Alon and Spencer for more.)

Line graphs are claw-free. And the random graph $G_{n, p}$ has in expectation $\Theta (n^4 p^3 (1-p)^3)$ claws (i.e., it's roughly some constant multiple of this). Having claws is not a monotone property, so we don't expect a classical threshold for when (as a function of $p$) we should think there ought to be claws. But nonetheless, a standard application of the second-moment method would tell you for what $p$ this property will hold almost surely (i.e., with probability tending to $1$ as $n$ tends to infinity). I did not check the following, but it's right up to some logs that I'm not writing:

The analysis splits into three regimes based on $p$.

If $n^{-4/3} \ll p \ll 1-n^{-4/3}$, then the number of claws will be concentrated about its mean, and the graph will have a claw with probability tending to $1$. Hence, the graph will fail to be a line graph almost surely.

If $p$ (resp $1-p$) is approximately of the form $C n^{-4/3}$, then the graph (or its complement) will be very sparse. I would guess that the number of claws would follow a Poison distribution, so it will be claw-free with probability tending to some $f(C)$. In this range, the graph (or its complement) will be extremely sparse, and in fact being claw-free would almost surely coincide with the property of being a disjoint union of paths (or its complement). Thus, in this range, $G_{n,p}$ (or its complement) will be a line graph with probability $f(C)$.

If $p \ll n^{-4/3}$ (resp. $1-p \ll n^{-4/3}$) then $G_{n, p}$ (resp. its complement) will almost surely be a disjoint union of paths, and hence it will be a line graph with probability tending to $1.$

[I'm being sloppy with the above treatment in the "critical window" when $p \approx n^{-4/3}.$ If you wanted to do this better, that's certainly possible.]

So in summary, unless $p$ is extremely small or extremely large, $G_{n,p}$ will almost surely fail to be a line graph. In particular, this holds for $p=1/2$, so almost all graphs fail to be line graphs.