  Magnetic materials with competing magnetocrystalline anisotropy and dipolar
energies can develop a wide range of domain patterns, including classical
stripe domains, domain branching, as well as topologically trivial and
non-trivial (skyrmionic) bubbles. We image the magnetic domain pattern of
Fe$_3$Sn$_2$ by magnetic force microscopy (MFM) and study its evolution due to
geometric confinement, magnetic fields, and their combination. In Fe$_3$Sn$_2$
lamellae thinner than 3 $\mu$m, we observe stripe domains whose size scales
with the square root of the lamella thickness, exhibiting classical Kittel
scaling. Magnetic fields turn these stripes into a highly disordered bubble
lattice, where the bubble size also obeys Kittel scaling. Complementary
micromagnetic simulations quantitatively capture the magnetic field and
geometry dependence of the magnetic patterns, reveal strong reconstructions of
the patterns between the surface and the core of the lamellae, and identify the
observed bubbles as skyrmionic bubbles. Our results imply that geometrical
confinement together with competing magnetic interactions can provide a path to
fine-tune and stabilize different types of topologically trivial and
non-trivial spin structures in centrosymmetric magnets.
