We use a "wet drawer", generally a shallow drawer next to the sink that is lined with a stainless or plastic liner. Everything damp, sponges, scouring pads, dish cloths, basket strainer, disposer stopper, etc. goes into the drawer.

There is enough ventilation to allow the items to dry, they are out of site and off the counter top.

The liner is easily removable for cleaning, and can usually be popped into the dishwasher.

I use little steel wool pads and toss them.
I gave up on sponges and the like - and bought some WS microfiber pads and toss into the laundry after use.

I keep the bottle washers, dish brush and similar in the drawer next to the sink, above the trash pullout. They go in the DW after use and I put them away dry. It also has a drying pad and the Vacu-Vin. I also put sponges and scrubbies in the DW, but since they get used many times a day, I keep them out, in a little ceramic sponge holder that matches the counter. It'll hold a large sponge, but I use a small one, a separate flat scrubbie and a curly corn scrubber, and those all fit. :)

In my old kitchen, I had a small sponge holder that suction cupped to the side of the sink, because I didn't have much room for anything else. It worked really well, kept the sponge to hand where it's needed (no way I'm searching in a drawer when I need a sponge now!), and kept it out of the line of sight.

Feisty- do you remember where you saw that basket that fit over the trash can? Only DH & I in our home, so I don't use the second trash can in my double pullout trash. That basket would be perfect for me.

I kept a tote under the sink for keeping cleaning supplies together, and when not in use I stored bottle brushes and the like there. Sponges I keep in a cute dish beside the sink ("yes, someone actually lives here!") and sometimes to a washable dishrag.

I dislike more draping a wet cloth over the sink divider than keeping a sponge in a cute dish. I wash synthetic sponges in the DW but the natural ones I just throw away, hopefully before they start to get gross.

Romy, I think it was an IKEA wire pullout drawer like this (they come in various widths) -

Here is a link that might be useful: RATIONELL Wire basket

Here's a photo of my under-sink area. I use a small ventilated hanging shelf from Ikea that is suspended from a curtain rod mounted inside the cabinet.

Fredanj - that is a brilliant under the sink idea!

Lol grrat idea! Now is the white bucket for catching drips?

Love, love, love that idea. I just had to print that picture out so I can put it in the folder with my other "good ideas" . Even if I don't do the basket, I love how you can hang those spray bottles on the curtain rod.

fredanj-
I showed your pic to my cabinet designer for my undersink area. I had saved it to an idea book months ago- I'm glad you posted it again so I can give you some credit!

I think the bar will be really handy to hang my kitchen towel/microfiber cloth and the spray bottles that I'm always needing access to!

Thanks for also sharing about that hanging basket.