The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian Aα=(i∇+A)2+αδΣ in L2(R2) with a δ-potential supported on a finite C1,1-smooth curve Σ are studied. Here A=12B(−x2,x1)T is the vector potential, B>0 is the strength of the homogeneous magnetic field, and α∈L∞(Σ) is a position-dependent real coefficient modeling the strength of the singular interaction on the curve Σ. After a general discussion of the qualitative spectral properties of Aα and its resolvent, one of the main objectives in the present paper is a local spectral analysis of Aα near the Landau levels B(2q+1), q∈N0. Under various conditions on α, it is shown that the perturbation smears the Landau levels into eigenvalue clusters, and the accumulation rate of the eigenvalues within these clusters is determined in terms of the capacity of the support of α. Furthermore, the use of Landau Hamiltonians with δ-perturbations as model operators for more realistic quantum systems is justified by showing that Aα can be approximated in the norm resolvent sense by a family of Landau Hamiltonians with suitably scaled regular potentials.