Basic investigations require the detectives' full attention, so a new investigation only becomes available once the previous investigation is finished. However, you can take part in special or additional investigations at the same time as the basic investigation.
Additional investigations become available when an informer gets a new lead. When this happens, an exclamation mark symbol appears near the informer's card, or a new quest icon appears. Some stages of additional investigations only become available once you reach a certain level.
Special investigations are connected to seasonal events, so they become available on specific dates.