Typhas and Phragmites are the main species used in constructed wetland due to their effectiveness, even though they can be invasive outside their native range.
In North America, cattails (Typha latifolia) are common in constructed wetlands because of their widespread abundance, ability to grow at different water depths, ease of transport and transplantation, and broad tolerance of water composition (including pH, salinity, dissolved oxygen and contaminant concentrations). Elsewhere, Common Reed (Phragmites australis) are common (both in blackwater treatment but also in greywater treatment systems to purify wastewater).
Plants are usually indigenous in that location for ecological reasons and optimum workings.