The A580 has many varied modes, and the autofocus types are dependent on which mode you are in.
Using the optical viewfinder, you have phase detect autofocus. This mode is as fast as can be and like any normal DSLR.
Using the 'quick' live view mode, you have phase detect autofocus. This mode is equally fast as with OVF mode, something completely unique to Sony DSLRs - no other DSLR can do this.
Using the main sensor live view, you can use phase detect autofocus, or contrast-detect autofocus, depending on your preference. Phase detect will focus faster, but in this mode will have to flip the mirror down and up to focus then restore the live view...with contrast detect AF it will be slower, but tends to be very accurate and will not have to flip the mirror to achieve the autofocus.
Sony DSLR-A580 / Sony 18-250mm / Minolta 50mm F1.7 / Sigma 30mm F1.4 / Tamron 10-24mm / Tamron 150-600mm / Tamron 90mm F2.8 macro / Minolta 300mm F4 APO
Sony A6000 / 18-55mm F3.5-5.6 / 55-210mm F4-6.3 / 10-18mm F4 / 35mm F1.8 / 16mm F2.8 / via manual adapter, lots of Pentax K mount, Konica K/AR mount, and Leica M mount manual lenses