I've always wanted a place that surveys average "wall voltages" across the US. It'll vary by neighborhood and even individual home. Heck, it'll vary based upon what you've got turned on due to voltage drop across the feed going from the utility company's transformer to your house.
120V is generally accepted by designers to be the current "standard" in the USA. The standard US voltage has been creeping up over the past 50 years or so. I guess it's been determined to be infeasible to upgrade everybody to 240V, so they just keep creeping what we have up, allowing the design targets to shift over the years. Hence, it's not uncommon to see 117V (the accepted "standard" about 15-20 years ago), 115V (~20-30 years ago), 112-113V (~30-40 years ago), or 110V (about 50 years ago) depending on the age of the service equipment (NOT necessarily dwelling age). Generally, designers expect at LEAST 5% variability and usually try to make it run off as wide of a range as they can.
Many electronics are designed for 100-120V range as 100V is line spec in Japan. Note that Japan uses wall plugs compatible with those used in the USA, though the third prong (earth ground) is very uncommon and even polarized outlets are only present in relatively new structures.
I gather you guys in Europe managed to standardize on 230V throughout the EU, including the UK which had to actually reduce their line voltage (which was up to like 237 or something before that standard was adopted).
If your residence is running unusually high (>125V) and it concerns you, you can usually get your utility to knock it down some. The transformers at the street that feed residences (usually one or two) have several output taps to facilitate this. Just call up and say you noticed that light bulbs were burning out quickly and the voltage measures high. Don't be surprised if they overcorrect slightly and knock you down to 115-117V.
Same if it's running really low (<110V). Call up and note that you are experience more noticeable than normal light flicker when your air conditioner turns on. Again, don't be surprised to end up with 120-125V if they do something.
FWIW, my house measures 121V (just measured with my assuredly accurate $5 multimeter) on an unloaded circuit with no major appliances running. It'll drop down to ~117V with lots of stuff in the house turned on or as low as 113-115V on a 15A circuit (AWG14) at near full load (e.g. vacuum cleaner or space heater). My friend's house in another city tends to run a little higher at ~123-125V unloaded.
Both of us have tons of stuff from Japan (stamped 100V); it all runs just fine. SMPS based designs frankly couldn't care in most cases, and most transformer based devices can tolerate some variability as it is expected in the design.
As for wiring your shed, I say spend the money to run a nice beefy (like AWG10) line with three conductors (plus ground, of course) to your shed. Put a sub-panel in your shed, and then you can run whatever you like. AWG10 should allow a 30A or maybe 40A panel (240V, of course, so 60-80A worth of 120V circuits). Consult all relevant codes, of course (I am not an electrician, and your local codes may vary, anyway). A sub-panel makes it easy to expand in case you want that oven, a welder, a large air compressor, etc.
If you want to go really big, mobile home feeder can be a relatively inexpensive medium to move a lot of juice, but it's often aluminum, not copper, so again consult all relevant codes regarding its use. Mobile home feeder is sometimes rated for direct burial, which can save you a conduit.