Enhanced ind-sheaves describe the Betti side of the irregular Riemann-Hilbert correspondence, in a manner compatible with Grothendieck's operations. In this way, classical constructions on the de Rham side have their natural topological counterpart. In this talk we will illustrate some examples in complex dimension one. In particular, irregular nearby and vanishing cycles, and their behavior under the Fourier transform. This is from joint works with Masaki Kashiwara.