Though not available for the shortwave bands, Google Play reviews are mostly positive for the new software defined radio application that will run on your Android phone or tablet: SDR Touch.
According to the Google Play page, SDR Touch covers 50 MHz to 2.2 GHz and demodulates WFM, AM, NFM, USB, LSB, DSB, CWU and CLW signals. It requires an inexpensive USB rtl-sdr compatible USB DVB-T tuner. Click here to search eBay for RTL-SDR receivers. You then connect to your Android device via a USB OTG cable and SDR Touch should control the receiver.
SDR Touch’s Google Play website has a list of supported receivers–make sure to check the model number from eBay against this list before purchasing. Additionally, you should install their free demo SDR app to make sure your Android device is compatible.
This video shows SDR Touch in action on the FM broadcast band:
SDR touch claims that crashes should be expected as this app is still considered somewhat experimental.
Would be great if SDR Touch could support the HF bands some day.