I've tried lots of different bags...right now I'm using a wicker fishing creel as a picking basket. I bought a real one from LL Bean that is actually meant for real use so is sturdy and has nice leather straps. It's pretty large so I can fit quite a few mushrooms in it.
I live in SE Alaska and do a lot of foraging in the woods around here. Terrain is pretty rough, lots of downed
trees in various stages of decay to scramble over, mossy floor often hiding ankle turning holes, patches of thick understory, wet, wet, wet so plenty of mucky areas, small streams to wade, rivulets to jump over, etc.
Good news is plenty of mushroom and berries. Bad news not always easy to get to.
I like to have my hands free when I'm making my way through all this. Plus when I do find the mushrooms it's nice to have my hands free for a cleaning brush or knife to trim up the stems. Species I'm collecting are often scattered so I'm picking one or two here, another one over there, walking a bit find one more, scramble into a blowdown to get to a nice patch, etc. This means it's nice to have something a I just drop a mushroom or two into and keep moving without having to constantly open and close a pack or swing it off my back. A traditional basket or bucket over the arm works sometimes but can often get in the way, caught up on bush, or ends up collecting large amounts of leaves, needle, twigs, and such as I'm scrambling around.
For ease of access and keeping my hands free I like to have a bag attached to my hip. I tried different style messenger bags but didn't fall in love with any. Soft construction style often meant the bag wouldn't stay open or squished the shrooms.
The fishing creel is nice in that it's wicker so doesn't smash. Being wet and rained on doesn't affect it (important around here), it lets the mushrooms breath, drip off, and drop spores as I'm collecting. I usually do have a few paper bags with me and I might use them to keep a mushroom separate (if I'm unsure of ID or don't want a fragile specimen to get broken). On longer trips I usually also carry a backpack for water, snacks, and supplies. If I find the motherload or a really huge bolete the overflow goes in paper bags in the backpack.
If I was hiking long distances, say to a remote burn site where I going to pick a large volume of say morels, and then have to pack them out I think I'd go with cloth or mesh bags in hand and when they're filled packed into a packbasket for the hike out. In easier areas with nice trails, open understory, or not traveling much distance I'd probably sling a wicker basket or small bucket over my arm.