I have a 2005 Aveo 5-speed, which has been my daily driver since I bought it new. I have absolutely no doubt that it's the worst handling vehicle in wet and especially snow that I've ever owned (driving for 45 years). I'm a skiier, and have driven many times in snow over the years, and have never been uncomfortable about driving snow-coverd roads. But with my Aveo, I noticed fairly early on that it didn't have the solid feel on wet pavement as compared to most everything else I've ever driven. Then I found it to be awful in the snow even with good all-season radials, so I bought separate rims and snow tires because I knew this was going to be an annual thing and didn't want to remount tires twice a year. With an aggressive snow tire, I consider it to be just adequate in snow, basically about average compared to everything else I've driven (but all of them with no snow tires required). Howerver, the odd thing is that it has always been a decent handling car on dry pavement. I have no idea what the wet/snow thing is all about, but it is what it is.
That being said, I'm not at all unhappy with the Aveo. The price was great, mpg also great, and repairs below average for me so far. So all things considered, it's been a very good vehicle. I believe that 2004-06, is basically the same, but have no idea if they improved the handling on wet/snow in later models. JMHO - others may offer different opinions.