It’s kind of a wide question, so I’d like to motivate it: When trying to determine the Hausdorff dimension of a set, I found that it is often reasonably easy to find it heuristically, but very difficult to actually prove it.
For example, we can split the Cantor set $\mathcal C$ into two parts $ \mathcal{C} \cap [0;\frac13] $ and $ \mathcal{C} \cap [\frac23; 1] $ which are just copies of $\mathcal C$, scaled by a factor of $\frac13$. Thus we have
$$ \mathcal{H}^s(\mathcal{C}) = \mathcal{H}^s(\frac13 \mathcal{C}) + \mathcal{H}^s(\frac 13 \mathcal{C}) = \frac{2}{3^s} \mathcal{H}^s({\mathcal{C})} $$
for all $s$. Now, if we assume that there is an $s$ with $\mathcal{H}^s(\mathcal{C})$ finite and non-zero, we can derive from this, that $s = \frac{\log 2}{\log 3}$ is the Hausdorff dimension of $\mathcal C$, by dividing both sides by $\mathcal{H}^s(\mathcal{C})$ and solving for $s$.
I found it quite difficult to prove this assumption—which, of course, doesn’t hold for any set—but I wonder if there are some criteria on which one can determine if such an $s$ exists. As discussed in this question, it is always true for Cantor sets, but I wonder if there are some nice criteria for fractals in general or even arbitrary sets.
$\newcommand\h{\mathcal{H}^s}$ Of course showing that $\h(E)=0$ (when true) is often no big deal; you can often simply write down an open cover that demonstrates this. But showing that $\h(E)>0$ directly from the definition is typically not so simple. Frostman's Lemma is often very useful here.
For example if $C$ is the Cantor set then the "Cantor measure" on $C$ shows that $\h(C)>0$ for $s\le\log(2)/\log(3)=\alpha$. (And it's then clear that $\h(C)=\infty$ for all $s<\alpha$, since $\mathcal H^\alpha(C)>0$.) Again, showing that $\h(C)=0$ for $s>\alpha$ is trivial from the definition.)