I used a medium handbag most of the time – folding changing mat (mine was a bambino mio one, it folds quite small), 1-2 reusable nappies (plus two emergency disposables at the bottom), wet bag, handful reusable wipes, tiny tube/tub of nappy cream, tiny bottle of water for wetting the wipes, change of clothes for baby, muslin. My phone and keys also fit in it, and I could fit a snack and small toy or two in as well.
If going out for longer (so more nappies) or somewhere I needed more stuff (more toys, more weather-dependent changes of outfit, etc.), then I used a rucksack we had already. Also used the rucksack before we got DD2's reflux under control, when I never left the house without 3 full outfit changes for her, one full change for me, and many, many muslins.