How bad is handling with only front aero wheel? I think it's obvious that the only reason to have aero wheel at front and regular wheel at rear is budget. I did some research on this and what I found was aero front wheel is more important and rear aero wheel is only to help bike handling despite the fact that most people use deeper rear wheels. Most stuff I read were from > years ago, wonder if anything changed (newer wheels handle better in crosswind maybe, etc.)

Edit
To clarify, some information that I found claim that using front only aero wheel results in worse handling than having front and rear aero.

So, can anyone confirm this and maybe someone with experience in using such setup. I want to know how bad handling can be with 50 mm front wheel and regular wheel at the back.
First, running an aero wheel in the back does help with drag, just not as much as the front. A 50mm section aero wheel in the front with a standard wheel in the back will be fine. It may look a little funny, but it won't make the bike unridable. That said, a deep section rim is a deep section rim, and it will cause some deflection from your course in crosswinds.

EDIT: in the case of a windy day and front only aero wheel, your front end is going to get pushed around more than your back. If you're running both front and rear aero wheels, you're going to get pushed around more evenly, but you're going to get pushed further. Neither situation is pleasant, especially if you're using clip-ons, but I don't think that going with only a front aero wheel is necessarily worse. It is a setup I've used in the past, and a setup that I think isn't altogether uncommon in lower categories as racers are more likely to borrow just a front from a fellow racer for a cat 5 time trial. If you're dead set on getting an aero wheel I wouldn't let potential handling differences vs a full set stop you.