Second-order half-linear differential equation (H): \({(\Phi (y'))'+f(x)\Phi (y)=0}\) on the finite interval I = (0,1] will be studied, where \({\Phi (u)=|u|^{p-2}u}\) , p > 1 and the coefficient f(x) > 0 on I, \({f\in C^{2}((0,1])}\) , and \({\lim_{x\rightarrow0}f(x)=\infty }\) . In case when p = 2, the equation (H) reduces to the harmonic oscillator equation (P): y′′ + f(x)y = 0. In this paper, we study the oscillations of solutions of (H) with special attention to some geometric and fractal properties of the graph \({G(y)=\{(x,y(x)):0\leq x\leq 1\}\subseteq {\bf {R}}^{2}}\) . We establish integral criteria necessary and sufficient for oscillatory solutions with graphs having finite and infinite arclength. In case when \({f(x)\sim \lambda x^{-\alpha}}\), λ > 0, α > p, we also determine the fractal dimension of the graph G(y) of the solution y(x). Finally, we study the L p nonintegrability of the derivative of all solutions of the equation (H).