  We analyze the gamma-ray sky at energies of 0.5 to 50 GeV using the
undecimated wavelet transform on the sphere. Focusing on the inner $60^{\circ}
\times 60^{\circ}$ of the sky, we identify and characterize four separate
residuals beyond the expected Milky Way diffuse emission. We detect the
\textit{Fermi} Bubbles, finding compelling evidence that they are diffuse in
nature and contain very little small-scale structure. We detect the "cocoon"
inside the Southern Bubble, and we also identify its northern counterpart above
2 GeV. The Northern Cocoon lies along the same axis but is $\sim 30 \%$ dimmer
than the southern one. We characterize the Galactic center excess, which we
find extends up to $20^{\circ}$ in $|b|$. At latitudes $|b| \leq 5^{\circ}$ we
find evidence for power in small angular scales that could be the result of
point-source contributions, but for $|b| \geq 5^{\circ}$ the Galactic center
excess is dominantly diffuse in its nature. Our findings show that either the
Galactic center excess and {\it Fermi} Bubbles connect smoothly or that the
Bubbles brighten significantly below $15^\circ$ in latitude. We find that the
Galactic center excess appears off-center by a few degrees towards negative
$\ell$. Additionally, we find and characterize two emissions along the Galactic
disk centered at $\ell \simeq +25^{\circ}$ and $-20^{\circ}$. These emissions
are significantly more elongated along the Galactic disk than the Galactic
center excess.
