They typically end up in archives, stored and catalogued then buried in a box on a shelf until someone needs to look at them. But do keep in mind even the mundane objects have value; nothing is boring about them. The public might not identify with them or the curator might decide the artifact doesn’t fit the narrative the museum is trying to convey but its never boring.

Some artifacts get stored and displayed rarely; could be to protect them from additional damage and it could be because the item isn’t relevant or interesting enough for the limited exhibition space available. Most institutions collections are expansive and are typically far larger than the exhibition space. Its still curated and stored and is often available upon request (usually have to prove your request to legitimate and not an attempt to steal or destroy it.)

If you found a gold button, a historical one with old-world decorations on it, and you lived in America you would likely be fine to keep it. If you found an arrowhead its likely all the museums would look it over and let you keep it. If you found the button in the UK you’d be expected to report it for evaluation; if found historically significant the state takes it and pays you gold spot price (if you’re lucky.) If you found a cache of arrowheads, tools and things that indicated a burial site or campsite you’d be expected to report it to one of the local institutions so it can be studied properly.

Theres a ton of variance in local laws about found artifacts so its hard to give a flat answer. In the states its so common to find ancient arrowheads that in most cases you’re allowed to keep them, heck lots of folks go hunting for them in freshly turned fields. In Britain its apparently common to find old roman coinage and it most cases the coins are worth a couple bucks and they let the individuals keep them. So institutions do turn away potential artifacts but even then I’d say the artifacts aren’t boring… just so common that better examples have been catalogued. Local laws always apply so metal-detectorists beware.

Feel free to delete this mods, as it doesn't have specific sources referenced. I am speaking as an archaeologist, and long story short it depends. The vast majority of collections in curation facilities and museums are exactly those "boring" pieces. When you dig a site, you're effectively destroying it. It's not like they're making any new 7,000 year old campsites. As archaeologists, we rely on the contextual associations between artifacts to make heads or tails of what people were doing. Towards that end, while we destroy the record/site, we take extensive documentation to make sure that context is recorded for researchers.

Part of the process after excavation is extensive cataloguing and curation of the materials and providing documentation about where the came from / how they were associated with other materials. We turn these notes and artifacts to museum for more or less permanent curation for future generations and researchers to have access to. Whether a museum takes a collection or not is dependent on a wide variety of factors. First and foremost is do you have an agreement with the facility to take materials? Part of budgeting for a dig is figuring out who will take the artifacts at the end, and to pay for their curation. It can be fairly pricey.

Next, the museum wants to know what are the legal and ethical considerations around these collections. Were they excavated by professionals with modern techniques, or were they excavated by an avocationalist and may or may not be well documented. Additionally, they want to know where they came from (e.g., private land, federal land, etc.) as there's numerous different laws around sites in these areas. Artifacts that were taken without context are often rejected, as the museum cannot be sure about the ethical acquisition of these materials.

Sometimes, collections are never looked at again. On the other hand, many collections are used and re-investigated for research projects in the future. Many archaeologists in academia are trying to focus on collections based research, and re-analyzing old sites using modern techniques. In the southeast for instance, the TVA built tons of dams, resulting in a flurry of salvage excavation. Many of these collections were quickly written up, but not subject to intensive analysis. Therefore, they represent a really valuable dataset to look at questions in the region about past lifeways and peoples.

Hope that helps answer your question! These "boring" artifacts are the backbone of understanding the past, and are critical to protect and curate for future generations and researchers. Feel free to ask any follow ups and I'll try to answer. The American Alliance of Museums has a fairly comprehensive page on ethics if you're interested in that.

https://www.aam-us.org/programs/ethics-standards-and-professional-practices/

So it can vary based on the institution and some other factors.

One of the most obvious factors is "does this pertain to what's already in the museum?". Obviously a museum that focuses on World War Two isn't going to accept a clay pot from Ancient Greece. In situations like that, its an clear answer of no.

However, other factors can kind of complicate things for other museums. A thing a lot of people dont understand, especially with big institutions like the Smithsonian Museum in D.C., is they have thousands of artifacts locked away in vaults that will never be seen by the public or maybe will only be displayed once every couple decades based on what instillation they are building. Some of these museums maybe only have 5-10% of what they have in their collections on display. A multitude of factors contribute to what's is chosen to be on display. They cant display everything because there simply isn't enough room and they dont want it to be just a cluttered mess of items everywhere.

Another factor is museums often run on slim financial margins and need people coming in with ticket sales/donations to help fund them. This means putting stuff on display that people want to see. Not everyone is going to be wowed by a clay plate or a plain looking spear. Stuff like machine guns from WW2 or maybe parts of a space rocket from the Apollo program are things that interest more people, hence why they are on display. Museums are kind of forced to "advertise" themselves towards the mainstream and the most common denominator to attract people in the door. This unfortunately means some stuff is left out as it is too "mundane"

So what happens with these other artifacts? Well, like I said most just sit in vaults. They are often used for reference collections and available for researchers to view and analyze. A lot of museums are working on digitization in order to make these items more accessible to the wider population for viewing on their websites or through online archives. As for will the general museum just take any artifact given to them, it depends a lot of things like available space, funding or money available, condition, ect. They wont just take any battlefield relic if its just falling apart because it just takes up space, has little to no value left in it, and the required upkeep to maintain and preserve it might just be too much for them to handle. Another factor is whether or not its a single object or if someone is trying to off load a large collection on them. A single item is going to be a lot less work to process as each item taken in needs to be authenticated, cataloged in some way so they know they have it, and then any preservation work that needs to be done to it. Large collections take the valuable time of museum staff to go through. Lacking in funding for staff to do these jobs can mean museums turn away more boring or run of the mill items in order to prioritize what they have or more "appealing" objects.

A lot of these objects can end up sold on the collectors market and find loving homes that way. Collectors, such as myself, love to take even mundane objects most of the time usually have more time we can devote to these objects over large institutions.

A great example of a museum that strives to put as much on display is the Cody Firearms Museum, which is part of the Buffalo Bill center in Cody Wyoming. A couple years ago they did a massive remodel and went from sort of the typical museum with stuff on the wall, to a more modern take. They have a large selection of their inventory now in these movable pieces that can be grabbed and dragged out from the wall. This both saves space but also gives a full 360 view of the many of the objects, a thing a lot of museums dont have. They have made it so now 80+% of their full inventory is now on display and not just sitting in the vault.