Micronations* are used commonly in the Numista catalogue to refer to 4 issuers (Lundy, Hutt River, Sealand and Seborga) that have their own issued "coinage" (usually listed as tokens), that are unrecognised by any sovereign nation or by the UN. Most of these issues are recorded in Krause's Unusual World Coins catalogue (X#); and while there are other "micronations" that have issued their own token coinage, as well as issues from the aforementioned 4 not in Krause, since they do not have X#, those "coins" are located in Tokens, or Exonumia on Numista.
Lundy

(above) Unofficial flag of the "Kingdom" of Lundy, never adopted (but used in the Numista catalogue)
Lundy is an island located in the Bristol Channel (a body of water between Cornwall/Devon and Wales). Celtic presence on the islands has been documented, and in the 12th century AD the Knights Templar supposedly owned it for a brief period of time. For the next few centuries it was a refugee from the law, with fugitives and pirates taking up residence there (in 1627, Barbary pirates even hoisted an Ottoman flag there), until the island was purchased privately in 1802, after which the island changed owners several times in 1834, 1918 and 1924, the last time going to a Martin Coles Herman, who proclaimed himself "King" of Lundy island.

(above) Location of Lundy Island
This wouldn't have been the first time the island acted as a separate de facto entity; in the 19th century colonies of Irish settlers there had supposedly issued their own "stamps" and "tokens". However those issues' existence is unconfirmed, as opposed to Martin Coles Herman's 1929 issues of stamps, and coins in denominations of half and one "Puffin" (after the island's puffin population), at par with the British halfpenny and penny. He was however charged and fined under the 1870 Coinage Act, and thus these "coins" never circulated, with the British pound sterling remaining the coinage used. After Herman's death in 1954, further fantasy issues were made in 1965 (for the 40th anniversary of his "becoming king"), in 1977* (for Elizabeth II's silver jubilee visit) and in 2011, with all these issues being of 1/2 and 1 puffin denominations (except the last, which also includes 2, 4 and 6 puffin issues).** All these coins bear the bust of Martin Coles Herman and some puffin motif on the obverse and reverse respectively.



(left) Common obverse (bust) of all Lundy issues (only difference being the date, either 1929, 1965, 1977 or 2011); (centre) Half puffin reverse (1929 original issue); (right) One puffin reverse (1929 original issue).
*It is unknown when the 1977 coins were struck; their first recorded appearance was only in 2014.
**The 2011 coins are listed in Exonumia, not under Lundy because they were completely unauthorised fantasy issues.
https://en.numista.com/catalogue/lundy-1.html
https://en.numista.com/catalogue/exonumia-1.html#devise3769
Hutt River

(above) Flag of Hutt River
The Principality of Hutt River is a micronation located inland near Perth, in Australia, that bills itself as the "second largest country on Australia". It was "formed" in 1970 when a disgruntled local farmer, Leonard Casely and some of his neighbours and friends declared a "secession" from Australia, citing wheat production quotas as a source of ire. The Governor of Western Australia threw his claims out, but Casely considers the government's "failure to respond" to be de facto autonomy from April 1972 onwards. In 1977 he was still prosecuted for failing to pay taxes, despite attempting to claim "sovereign immunity".

(above) Location of Hutt River
Over the next few decades the "Principality"'s status changed to a Kingdom, then back to a Principality, as well as issuing several stamps and coins (in dollars) since the 1976, most of which are commemorative souvenirs. Legal disputes with the Australian Government have occured in 1984 and 2006, when both times Casely lost (with arguments called "fatuous, frivolous and vexatious"). Notably, in June 2017, Casely (who calls himself "HM Prince Leopold I of Hutt") announced his abdication in favour of his youngest son*. They are still being involved in legal issues with the Australian government over unpaid taxes. Interestingly, the Principality's Latin motto "Dum Spiro Spero" is identical to Sarawak's.



(left) Bust of Prince Leonard I, depicted on the obverse of many early Hutt River issues. (centre) (right) Obverse and reverse of a Hutt River commemorative in the style of a US Kennedy half dollar, celebrating 42nd US President, William "Bill" Clinton.
*Hutt River's official website can be found here.
https://en.numista.com/catalogue/principality-of-hutt-river-1.html
Sealand

(above) Flag of Sealand
The Principality of Sealand is a micronation located on a former military offshore platform off the coast of England, in the North Sea, that is probably the most well-known micronation (as well as the smallest). The platform it is located on (called HM Fort Roughs) was originally built in 1943 as part of the Maunsell Forts, to help defend nearby shipping lanes from German planes during WWII. It saw little action over the last two years of the war, and was finally abandoned in 1956. It sat abandoned until 1965, when a couple of pirate radio broadcasters squatted there briefly; and then in 1967, Major Paddy Roy Bates (a UK citizen) arrived intending to set up his own pirate broadcasting radio system. However, he never actually got around to doing this, instead declaring the "independence" of the Principality of Sealand, and claiming the tower and the surrounding waters as it's sovereign territory that September.

(above) Location of Sealand
This led to a 1968 incident where Bates was tried for firing at British workmen attempting to service a nearby buoy; but he was acquitted after the court ruled "Sealand" was outside of British territorial waters. From 1975 onwards, coins and stamps (in dollars) were issued in the name of the Principality, the coins being purely commemorative issues. In 1978 a "diplomatic incident" occurred on Sealand when a German national, Alexander Achenbach (self-described "Prime Minister of Sealand") hired mercenaries to capture Sealand while Bates was away; his son Michael, however was there; he "defeated" the "invasion" and held Alexander hostage. The British government disavowed the incident, but Alexander was released a while later anyway; he continues to claim rightful ownership of Sealand, establishing a "Sealandic Rebel Government [in-exile]"*. Since 1987 Sealand is now in British territorial waters, but unlike Hutt River, has never attempted any legal disputes over sovereignty with the UK; it has no recognised legal status anywhere, and more recently, Sealand's "Prince" Roy Bates died in 2012. His son Michael continues to run Sealand to this day**.



(left) Bust of Roy Bates, from a 1977 commemorative Sealandic coin issue; (centre) 1994 commemorative "Orca - Treasures of the Sea" issue bearing Sealand's coat of arms; (right) "Rebel" issue by Achenbach's successor Johannes Seiger from 1991.
*The rebel Sealandic government in exile has it's website here (in German) where Seiger continues to claim to this day that he is the "rightful ruler" of Sealand.
**The Sealandic website here is where you can pay to become a Lord/Lady of Sealand.
https://en.numista.com/catalogue/sealand-1.html
Seborga

(above) Flag of Seborga
The Principality of Seborga (Principato di Seborga) is a micronation located in northwestern Italy, near the French border, in the small town of Seborga. It is interesting because unlike the others, an "actual" contender for a "real" Principality by that name existed in the past; at least this is the claim made by it's founder Giorgio Carbone. Supposedly the abbot of local Benedictine monks was made a prince of the Holy Roman Empire in 1054, and until a disputed sale in 1729 to the House of Savoy (of Sardinia-Piedmont), the Principality existed as a legal entity. The disputed status of the 1729 sale is cited by Giorgio, who also brings up the lack of subsequent mention of Seborga in the Congress of Vienna (1815) or the Act of Unification [of the Kingdom of Italy] (1861), and the exile of King Umberto II of the House of Savoy after WWII (1946). In the early 1960s, Giorgio began promoting the idea; and in 1963 the people of Seborga crowned him "His Serene Highness Giorgio I, Prince of Seborga".

(above) Location of Seborga
Unlike the others, the Italian government has not had any (public) legal disputes with Seborga. A 1995 referendum voted 304-4 in favour of the principality's independence from Italy, "reaffirming" the principality's "legitimacy". Coins were issued in the "Luigino" in 1995-96 and 2012-13, most bearing the bust of Prince Giorgio I and some variant of the Seborgan coat of arms. Recently in 2006 and 2016, contested claims to the throne of Seborga caused minor local controversy, with Giorgio himself passing away in 2009 and succeeded by Marcello Menegatto. Interestingly, the position of monarch is not hereditary and is actually elected every seven years, with Marcello defeating a British DJ in 2017 for the position. (No, that is not a joke.)



(left) (centre) 1 Luigino coin from 1996's obverse and reverse, bearing the bust of Giorgio I and the Seborgan coat of arms (right) 2012 commemorative, bimetallic issue commemorating the deceased Giorgio and new Prince Marcello.
https://en.numista.com/catalogue/seborga-1.html