1. Lovely: I have to admit that I am completely biased on this one because I do tend to judge an app by it's cover and this one is well-designed and, well, Lovely! With this free app, set up neighborhood alerts (down to the radius of a few blocks), learn which saved favorites are off the market, and email or phone directly from the listing.
2. Padmapper Apartment Search: This one is great. This app is detailed by neighborhood, and can show you for-rent housing ordered by distance from your current home, plus its filtering option helps you narrow down your search.
3. Rentals.com: This one was created by the well-known rental website of the same name. It has a full database of listings, including options to contact the manager directly from the app.
4. Real Estate by Trulia: This app uses a comprehensive layering technique. It can source for-rent apartments and houses nearby, then give you layers of information such as local boundaries, amenities, schools and other useful information.
5. Apartment Guide: Available for both iOS and Android users, this app taps into your device's GPS chip to pinpoint available apartments near your current location (or you can enter a location via the city, state or ZIP code). The apps is especially good for previewing property photos and floor plan so you can narrow down search results to what you're really looking for.
(Images: as linked above.)