Hows does the particular value of $p$ determine the functions that are excluded from $L^p$ spaces? What functions in general do the various $L^p$ spaces exclude? For example I've read that $f(x) = x^{-\frac{1}{2}}$ is not in $L^2(D)$ where $D = (0,1)$. So it seems that there is a direct relation between the inverse of the exponent and the $p$ value of the space?
Is it the case that in general $f(x) = x^{-\frac{1}{p}}$ is not in $L^p(D)$? Is that all there is to $L^p$ spaces or are there more complicated functions that are more difficult to classify as belonging to some $L^p$ space? Do the properties of the domain make a difference (dimension, closed, compact, etc...)?
Essentially what specific type of functions get excluded when we define a function $f$ and a value of $p$ such that $f \in L^p$? Some explicit examples (such as $f(x) = x^{-\frac{1}{2}}$ would be particularly helpful to get some intuition on these spaces?

Why don't you calculate it yourself:  For which exponent $a$ is $x^a$ square-integrable on $(0,1)$?  The same for $p$-integrable?
Other example:  For which $a,b$ is $x^a (\log x)^b$ in $L^p(0,1)$?
Examples on $L^p(\mathbb{R})$ appear more contrived, since $x^a$ is not in that space for any $a$.  (You either have too big of a singularity or have too fat tails at infinity).
There are lots of answers on MSE about this already: $L^p$-space inclusions
