We introduce an online version of the emph{multiselection} problem, in which
$q$ selection queries are requested on an unsorted array of $n$~elements. We
provide the first online algorithm that is $1$-competitive with offline
algorithm proposed by Kaligosietal [ICALP 2005] in terms of comparison
complexity. Our algorithm also supports online emph{search} queries
efficiently. 

We then extend our algorithm to the dynamic setting, while retaining online
functionality, by supporting arbitrary emph{insertions} and emph{deletions} on
the array. Assuming that the insertion of an element is immediately preceded by
a search for that element, we show that our dynamic online algorithm performs
an optimal number of comparisons, up to lower order terms and an
additive~$O(n)$ term. 

For the external memory model, we describe the first online multiselection
algorithm that is $O(1)$-competitive. This result improves upon the work of
Sibeyn [Journal of Algorithms 2006] when $q \gt m$, where $m$ is the number of
blocks that can be stored in main memory. We also extend it to support
searches, insertions, and deletions of elements efficiently.