In 1981 Samuels and Steele showed that the maximum expected length of subsequence which can be selected from $n$ iid observations by a nonanticipating online strategy is asymptotic to $\sqrt{2n}$. This can be compared with the classic $2\sqrt{n}$ asymptotics for the (offline) longest increasing subsequence in the Ulam-Hammersley problem. The talk will survey history of the online selection problem and then focus on recent results on its poissonised version. The latter include a renewal-style approximation leading to an asymptotic expansion of the moments, as well as functional limits for the transversal and longitudinal fluctuations of the shape of the increasing subsequence under the optimal or near-optimal selection strategies. The talk is based on a joint work with Amirlan Seksenbayev.