As car-free newlywed grad students, my wife and I used to go grocery shopping with two huge framed backpacks, plus the Pletscher mousetrap racks on our bikes. If I were doing heavy grocery hauling today, I would try either baskets or panniers. For light-to-moderate grocery trips, I have a "Bike Bag," which I keep permanently attached to the Blackburn Mtn. Rack on my mountain bike. The bag, a $3 yard sale treasure, is superb for fragile or dense grocery or hardware items I would prefer not to carry in a backpack.
Shopping/utility duty is where a basic old 1970s 10-speed (if you have hills and traffic, as I do) or 3-speed can really shine. I have successfully used a Schwinn Varsity and a Peugeot UO-8 in this application. Lighter bikes, such as my 1959 Capo, can get pretty squirrelly when loaded even moderately.
The other very popular utility bike in my neighborhood is any basic balloon-tyred coaster-brake middleweight, with a surfboard rack. My neighbor relishes the freedom from wallet and car key hassles he obtains by bicycling instead of driving to Moonlight Beach, Grandview, Swami's, Stone Steps, or Beacon's.
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"Far and away the best prize that life offers is the chance to work hard at work worth doing." --Theodore Roosevelt
Capo: 1959 Modell Campagnolo, S/N 40324; 1960 Sieger (2), S/N 42624, 42597
Carlton: 1962 Franco Suisse, S/N K7911
Peugeot: 1970 UO-8, S/N 0010468
Bianchi: 1982 Campione d'Italia, S/N 1.M9914
Schwinn: 1988 Project KOM-10, S/N F804069