I don't use filters

Let's take the physical filters first. It's a debate that's been going on for as long as I have been taking pictures, even more so back when I shot film, and I have experienced the discussion becoming very heated between pro- and anti-filter proselytes on many occasions.
Personally I'm neither anti or pro, but I simply just don't use filters – apart from a polarizer now and then and sometimes a neutral density filter. I use them to achieve an effect that can be difficult or impossible to achieve in other ways.

So this is not an array of arguments against using physical filters. Use them if you please or if they help you get better pictures. Or if you feel like it, or your dealer says it's the best thing you can do for your lenses. I just don't use filters such as UV or skylight because I can't see that they have any effect on my pictures. People can argue all they like that such filters make a difference, but I still can't see it in my images. Back in the film days there might have been a reason to use the so called haze filters, because UV light did sometimes cause blue haze on color film. But these days digital doesn't have the same sensitivity to UV light and will not have the blue tinting. The filter doesn't do much good.

On the other hand I don't see any bad effects either. Many anti-filter photographers argue that the filters create more reflections and deteriorate the image in some way – especially if the filter is low quality – which makes sense. Some see lower contrast, some see color fringing, some see more reflections and ghosting. But I have also failed to see these effects. I simply can't see any difference with compared to without a neutral filter.

I also hear the argument that the filter protects the lens. That might be the case, but I can say again that I have never needed that protection and never had an incident with a lens where a filter would have saved it. I have dropped lenses and dipped them in saltwater. I have fallen several times, camera first, but even when I had to scrap out a shovelful of sand from the front of my brand new Sigma 18-35mm f/1.8, there wasn't a single scratch on the lens. Another time the hood broke on my Nikon 17-55 f/2.8 and last time that lens took a hit, it broke clean off the D300s leaving the mount and screws in the camera! A filter would have done nothing to save me from damaged lenses and costly repairs.
It's worth noting that I'm not exactly careful with my gear, which is exposed to sand, dirt and water – saltwater in particular – and I usually don't do much to protect it from rain or dust except where these elements are torrential or have a negative influence on the images. I occasionally see sprays of water and dirt on the front element of my lenses, and usually just carefully clean it off with a blower and fiber cloth made for the purpose.
Most of the time I use the lens covers front and back when I pack down lenses for transport, but never have one on when I'm carrying the camera.
I own dozens of lenses and have owned many more, and have never had one, which had a scratch or any other kind of permanent blemish on any glass element. I bought one used lens once, which had fungus internally, but neither filters nor covers could have prevented that.
I have a couple of memorable experiences with filters, but not good ones.

I have a polarizer that came apart with the rotating part loosing connection with the part screwed into the lens. It is a Hoya filter, so the quality should be OK. I managed to get the thin, threaded part unharmed off the lens and get it assembled again, and it still works today.

My other experience was slightly more unnerving: a large UV filter was very permanently stuck in the front threads of a Tokina 100-300mm f/4 that I bought used. I could probably have left it there with no ill effect, but decided to get it off. No matter what smart tricks I tried, the filter didn't budge. Finally I removed the thin metal ring that held the glass in the filter ring and removed the glass. After that I could simply clip the final metal ring with a pair of clippers, and peel the threaded ring out by bending and breaking it. It was almost chemically fused with the lens! Needless to say that I haven't screwed a filter on that lens since – or any other lens for that matter.
I wasn't an avid filter user before this experience, and the incident certainly didn't increase my use. Whenever I do use a filter I screw it very gently into the front threads on the lens, and unscrew it as soon as I have shot. These days I simply have no filters mounted on any of my lenses, and routinely screw off the ones I get on used lenses and politely turn down the standard offer from all photo salesmen who seem to get a kickback (and a personal kick) from every filter they sell judging from their eagerness to add a filter to every lens sale.
I use neutral density filters now and then, and have a filter holder, which screws into the front of my lenses using on of a set of adapters. I have also used a small matte box for mounting square filters of different kinds. The box uses the same type of screw in adapters as the filter holder. It's really not often that I use any of them.

I'm not a great fan of digital filters either. One click Instagram-style filters don't appeal the least to me. I find them tiring and not particularly creative. Fading colors, sepia tones or adding scratches and corner falloff is a bit bland, and many of the other filters are also too much, too banal or too overused.
My ancient Photoshop has a ton of filters, which can create all kinds of strange and familiar effects. I don't use them directly on photos, but sometimes use some of them in subtle ways in post processing. I do post process photos in ways that can be called filtering. I often use the classical filters when converting to B/W where blue, red and yellow filters can give a specific tone and contrast to an image. I have also fooled around with the Nik effects packages, and I currently have DxO Filmpack 3 installed. I wouldn't call any of these packages filters as such, but they do of course have certain traits in common with Instagram filters or filters in Photoshop.
I have played with the Orton effect, which could also be called a filter. I used it once for one project, but haven't really used it since.

I'm simply not a big fan of filters. End of story.

This excellent article by Steve Perry strongly supports my empiric findings and certainly doesn't make me want to use filters more than I do now.