Organized growth of high aspect-ratio nanostructures such as membranes is interesting for opto-electronic and energy harvesting applications. Recently, we reported a new form of InAs nano-membranes grown on Si substrates with enhanced light scattering properties. In this paper we study how to tune the morphology of the membranes by changing the growth conditions. We examine the role of the V/III ratio, substrate temperature, mask opening size and inter-hole distances in determining the size and shape of the structures. Our results show that the nano-membranes form by a combination of the growth mechanisms of nanowires and the Stranski-Krastanov type of quantum dots: in analogy with nanowires, the length of the membranes strongly depends on the growth temperature and the V/III ratio; the inter-hole distance of the sample determines two different growth regimes: competitive growth for small distances and an independent regime for larger distances. Conversely, and similarly to quantum dots, the width of the nano-membranes increases with the growth temperature and does not exhibit dependence on the V/III ratio. These results constitute an important step towards achieving rational design of high aspect-ratio nanostructures.