The Common Locale Data Repository is an excellent resource for locale data. From the website you can download an xml version of the database, which includes datetime formats, number formats, and lots of other locale specific data.

You can get it programmatically like this:

//by culture
var formats = CultureInfo.GetCultures(CultureTypes.AllCultures)
    .ToDictionary(x => x.Name, x => x.DateTimeFormat.GetAllDateTimePatterns());

//or by country (only specific cultures has region/country info)
var formats = CultureInfo.GetCultures(CultureTypes.SpecificCultures)
    .GroupBy(x => new RegionInfo(x.Name).DisplayName)
    .ToDictionary(x => x.Key, x => x.SelectMany(y => y.DateTimeFormat.GetAllDateTimePatterns()).Distinct().ToArray());

For reference, this is the most comprehensive I've found

This function returns a string containing formatting data for the current date/time, based on the provided locale string.

function getDateFormatString(locale) {
    
  const options = {
    hour: "numeric",
    minute: "numeric",
    second: "numeric",
    day: "numeric",
    month: "numeric",
    year: "numeric",
  };

  const formatObj = new Intl.DateTimeFormat(locale, options).formatToParts(
    Date.now()
  );

  return formatObj
    .map((obj) => {
      switch (obj.type) {
        case "hour":
          return "HH";
        case "minute":
          return "MM";
        case "second":
          return "SS";
        case "day":
          return "DD";
        case "month":
          return "MM";
        case "year":
          return "YYYY";
        default:
          return obj.value;
      }
    })
    .join("");
}

console.log(getDateFormatString("en-US")); //Expected Output: "MM/DD/YYYY, HH:MM:SS PM"

console.log(getDateFormatString("ko-KR")); //Expected Output: "YYYY. MM. DD. 오후 HH:MM:SS"

Edit options to change the data extracted from the date object and edit/add cases to accommodate it.

Find the documentation for DateTimeFormat here.
I adapted the function from an article on newbedev.com.