localeDatePattern turns a locale's D_FMT into a pattern, so the same date renders the way each culture writes it — month-first in the US, day-first in Germany (here march15 is 15 March 2020):

do us  <- localeByName "en_US.UTF-8"
   de  <- localeByName "de_DE.UTF-8"
   usP <- localeDatePattern us
   deP <- localeDatePattern de
   pure (format usP march15, format deP march15)       -- ("03/15/2020", "15.03.2020")

Use currentLocale instead of localeByName to follow the machine's own LC_TIME setting, and parse with the same pattern to read that layout back:

do loc <- currentLocale
   p   <- localeDatePattern loc                        -- the current locale's short-date layout
   pure (format p march15) >>= parse p                 -- round-trips in whatever order the locale uses

localeTimePattern does the same for the time-of-day layout (T_FMT), and localeDateTimePattern for the combined date-and-time layout (D_T_FMT) as a CalendarDateTime.

localeDateTimePattern deliberately ignores the time zone. Most D_T_FMT strings end with %Z (a zone abbreviation such as CEST) or %z (a numeric offset); a CalendarDateTime is civil time with no zone attached, so there is nothing to render there and nothing to interpret, and those specifiers are dropped from the compiled pattern. In particular a zone-less datetime is not assumed to be UTC — treating civil time as UTC is exactly the accidental coupling the library is built to avoid; turning a CalendarDateTime into an absolute instant always requires you to attach an offset or time zone on purpose.

When you do want the zone, use parseZonedDateTime (below): it parses the locale's zoned layout into a ZonedDateTime, capturing the %Z abbreviation and resolving it through a provider you supply (abbreviations are ambiguous, so the caller owns that mapping). It requires the layout to contain a zone, throwing ZonelessLayoutException otherwise — a layout with no zone is not a zoned value.

A layout that instead carries a numeric offset (%z, e.g. +0200) is unambiguous, so localeOffsetDateTimePattern compiles it into an ordinary, pure, bidirectional OffsetDateTime pattern (used with parse and format, no provider needed), throwing OffsetlessLayoutException if the layout has no %z.