Etymology
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miner (n.)

c. 1300 (early 13c. as a surname), "one who mines, a person engaged in digging for metals or minerals or in digging a military mine," from Old French mineour (13c.), agent noun from miner "to mine" (see mine (v.1)).

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mining (n.)

1520s, "the business or work of a miner," verbal noun from mine (v.1). From c. 1300 as "the undermining of walls or towers in a military attack." Mining-camp "temporary settlement for mining purposes" is by 1853, in a California context.

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mine (v.2)

"lay explosives," 1620s, in reference to old tactic of tunneling under enemy fortifications to blow them up; a specialized sense of mine (v.1) via a sense of "dig under foundations to undermine them" (late 14c.), and miner in this sense is attested from late 13c. Related: Mined; mining.

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Stakhanovite (n.)

1935, from name of hard-working Soviet coal miner Aleksei Grigorevich Stakhanov (1906-1977), in reference to an efficiency system in which workers increase their piecework production and are rewarded with bonuses and privileges. Soviet authorities publicized his prodigious output as part of a campaign to increase productivity.

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mine (v.1)

c. 1300, minen, "to dig a tunnel under fortifications to overthrow them," from mine (n.1) or from Old French miner "to dig, mine; exterminate," from the French noun. From mid-14c. as "to dig in the earth" (in order to obtain minerals, treasure, etc.). Figurative meaning "ruin or destroy by slow or secret methods" is from mid-14c. Transitive sense of "to extract by mining" is from late 14c. For the sense of "to lay (explosive) mines," see mine (v.2). Related: Mined; mining.

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axe (n.)

"edged instrument for hewing timber and chopping wood," also a battle weapon, Old English æces (Northumbrian acas) "axe, pickaxe, hatchet," later æx, from Proto-Germanic *akusjo (source also of Old Saxon accus, Old Norse ex, Old Frisian axe, German Axt, Gothic aqizi), from PIE *agw(e)si- "axe" (source also of Greek axine, Latin ascia).

The spelling ax is better on every ground, of etymology, phonology, and analogy, than axe, which became prevalent during the 19th century; but it is now disused in Britain. [OED]
The spelling ax, though "better on every ground, of etymology, phonology, & analogy" (OED), is so strange to 20th-c. eyes that it suggests pedantry & is unlikely to be restored. [Fowler] 

The meaning "musical instrument" is 1955, originally jazz slang for the saxophone; rock slang for "guitar" dates to 1967.

To have an axe to grind is from a Sept. 7, 1810, essay in the Luzerne (Pennsylvania) "Gleaner" by U.S. editor and politician Charles Miner (1780-1865) in which a man flatters a boy and gets him to do the chore of axe-grinding for him, then leaves without offering thanks or recompense. It was published in a collection in 1815 titled "Essays From the Desk of Poor Robert the Scribe." The story ("Who'll Turn the Grindstone?") has been misattributed since late 19c. to Benjamin Franklin, a mistake continued in Weekley, OED print edition, "Century Dictionary," and many other sources (Bartlett's "Familiar Quotations" has gotten it right since 1870).

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