If it does matter, what is the current state of the art elliptic curve and how does it compare with popular elliptic curves such as Curve25519 or secp256k1?
Well, if you have an elliptic curve with a large subgroup of size $q$ (which is prime), then we know how to compute a discrete log within that subgroup in $O(\sqrt{q})$ time, and this applies to all elliptic curves (actually, all groups).
So, to make this attack take $2^{128}$ time, we need a $q \approx 2^{256}$.
And, because of the Hasse theorem, for a prime curve of characteristic $p$, we have $p + 2\sqrt{p} > q$, or in other words, the smallest $p$ can be is about 256 bits.
The standard way to represent a public key is to give the $x$ coordinate as an integer; this is a value between 0 and $p-1$; that is, a 256 bit value.
Hence, selecting a curve other than Curve25519, secp256k1 or P256 doesn't buy us anything; either that alternative curve would have reduced security or have a public key that's as least as large.
About the only thing you can try to come up with a reduced method of transmitting the $x$ coordinate; one simple-minded approach would be to always select an $x$ coordinate with $k$ bits of 0 at the top (and just not transmit those $k$ bits explicitly); finding such a key using rejection sampling would take $O(2^k)$ time and would save $k$ bits - perhaps doable if you need to save a byte or two - obviously infeasible to save more than that. I don't know of a cleverer approach to find public keys that meet a similar space saving technique.
