Does skin effect appear in home wires?
Skin depth is given by :
$$
\delta = \sqrt{\frac{1}{\sigma \pi \mu f}}
$$
With in this case (approximate values) :
$$
\sigma = 6.10^7\text{ (copper's conductivity)}\\
\mu = \mu_0 = 4\pi 10^{-7}\\
f = 50 Hz.
$$

Which gives
$$
\delta = 9.2 mm
$$

So skin effect would be observed on cables with radius greater than 9.2 mm which means 18.4 mm diameter (3/4 inch) (or a 266 m² cross-section), which is very close to AWG(7\0), a gauge qualified to carry hundreds of amperes (while a domestic installation carries no more than 50 A). 
Since most cables for home are less than that, the skin effect is often and justifiably neglected in home electricity applications.

Also, the skin depth is the distance from the surface of the conductor to where the current density is damped to less than 1/e (36%) of its surface value. So in fact you would need even more diameter to observe the center of the cable having no current.

Regarding the choice between stranded and solid cables, solid cables are often preferred for home wiring for different reasons including cost, size, durability. Those have already been discussed in here :
https://diy.stackexchange.com/questions/5901/why-are-homes-wired-using-solid-wire-rather-than-stranded