Now that we know that the universe is expanding, how much does a linear lightyear of space expand currently in a year?
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A recent estimation of the Hubble constant is 71.9 (km/s)/Mpc.
This means there is an expansion of
$$\left(71.9\ \mathrm{\frac{km}{s\ Mpc}}\right)\left(\mathrm{\frac{60\ s}{min}}\right)\left(\mathrm{\frac{60\ min}{hr}}\right)\left(\mathrm{\frac{24\ hr}{day}}\right)\left(\mathrm{\frac{365\ day}{yr}}\right) = 2,267,438,400\ \mathrm{\frac{km}{yr\ Mpc}}$$
To apply this expansion over a light year:
$$ \frac{2,267,438,400\ \mathrm{(km/yr)/Mpc}}{3,261,564\ \mathrm{lyr/Mpc}} = \boxed{695.2\ \mathrm{\frac{km/yr}{lyr}}} $$
which is about $7.3483\times10^{-11}\ \%$ per year or $432\ \mathrm{miles/year}$.
A linear light year of space expands by 695.2 km/yr with a given Hubble constant of 71.9 (km/s)/Mpc.
Edit: A better measurement released on 2018-02-22 using parallax found the Hubble constant to be 73.45 ± 1.66 (km/s)/Mpc.