My koga-miyata came with a small rack in the front, it's cute but it's not designed for panniers or anything heavy. (The back rack was flimsy too, but I replaced it when I built a stronger wheel this winter past.)
First, I thought about buying a cheap basket for the front to carry my purse. However, the baskets at the market are too short, so they hang from the bar, tilting into the brake cable that runs up the front tube.
Next, I looked at some classic attachable handlebar bags from Ortlieb and other brands, but I found them unappealing and not versatile enough. The black plastic clip that has to be installed on the handlebars is ugly and cannot be quickly removed (only the bag is quick to remove). Icky. Practical, but icky. Also, too highly specialized- putting my actual purse/bag in a bag just seems redundant somehow and I'm not going to walk around with a classic handlebar bag as a purse.
There are many "fashionable" handlebar bags out there, but they also seem unappealing, too small, or too girly... and from all the buckles and whatnot, I'm not sure how quickly they would come off the bike. Probably not as easily as a classic bag, but at least they don't leave behind a hunk of plastic.
Instead, my solution was to find a purse/bag designed as a purse/bag that could fit on bicycle handlebars. I needed a strong very adjustable strap, and a stable velcro flap system for enclosure. That was not hard to find! Many bags are designed this way... And it works perfectly on my bike. I can easily access whatever is inside, I can take the whole bag on and off in a few seconds, and it's a simple purse if I want to use it without the bicycle.
The strap goes under the handlebars and inside the top of the flap.
Then, tighten the strap and tuck the excess loop inside.
Close the velcro flap.
Photo by the Travelling Two!