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.
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