I have the same bike – an ’07 SV650 – and I recently changed my OEM Dunlop to Dunlop Road Smart, a sport touring tire. Lots of people on svrider.com like the Road Smart. Another tire I considered was the newly designed Perilli Angel ST, but they cost a little more. I have Bridgestone BT-021, which are a little cheaper, on my YZF600R, and they work pretty well. The bottom line is I feel all the major brand sport touring tires are more than good enough for general street riding, so it comes down to cost and personal preference.
Most people feel the new generation sport touring tires can handle anything you do on the street, even beginner track days, and they’ll last 8-10k miles. Sport touring tires may actually have the advantage in bad weather riding – including cold and rain. For people who ride very hard on the street and do more track days, a stickier sport tire, such as Michelin Pilot Power, would have more grip, but they may last only 3-5k miles.
Do a “tire” search on svrider.com and you’ll have a lot to read about.