"Rat" (council), "Rad" (wheel), "Bürger-Rathaus" (village center, mayor's office), "Fahrradhaus" (bicycle house), "Ratschlag" (hint, tip), "Radschlag" (wheel [gymnastics]). The name Conrad supposedly derives from Rath (Con either from coen "king" or from kühn "wise, courageous"; cp. Cohn, cone etc "yewish council, priest"?).
"Rand" (rim), I guess, gave "ranzig" (rancid).
"Bund" (binding [of flowers] union [of countries etc]) and "bunt" (colorful) are so hard to distinguish I'm not sure how to spell "Bun?metall" (wiki says colorful, but since Buntmetalle are ligaments [cf. Legierung] like bronce, abounding in a certain assortment of metalls, there's a mixup I'd say).
"Berg" (hill, mountain), "Werk" (work, product), "Bergwerk" (mine, shaft) but "bergen" (to unearth) "werken" (to work). g/k can be reduced to a glottal stop Ber' and Wer', loosing the distinction. Berg can be lenited to Berch, in one (my own hodgepotch) dialect, but not so for Werk. There's yet a difference between weich (soft) and Weigel (a name) that didn't go to *Weichel but rather *Weiyel (voiced velar fricative [ɣ̞]).
Berliner Ick, Icke seems to retain an old variant over ich (I, ego).
In gucken/kucken the standard couldn't overcome the dialect so kucken is regarded as standard alternative (or alternative standard?).
weg (way, long e) weg (away, short e)--A broad way and a quick away, so to speak--Zweck (purpose) Wecke (bread; southern dialect; surely Weck' somewhere where it's in use, also Weckerl').
Hagen (a name; a kind of place) hager (slim; cp. mager - meager, lean) Haken (hook) haken (to jam) hence poetic hag' and hak'.
mag (I may, like) mögen (to like) vermögen (to be able) mach (I make, do) machen (to make) machte (I did) möchte (i want, would like) -- it almost seems as if möchte was a subjunctive preterite(?) of machen (cp. might, Macht) and mögen a backformation when möchte were reanalyzed as its own verb. Additionally complicated by meiern (to lead, produce, set into motion), mayor and Lt. mega.
"Birken" (birch trees), "wirken" (to effect, work) shows that birch fortified (or vice-versa?). "Henkel" (cup-handel etc.) vs "hängen" (to hang), "Henker" (executer), similarly?
May overlap or clearly differenciated in g/k: Angst (angst, fear) dankst (2. p. sg. to thank) denkst (2. p. sg. to think) langst (2. p. sg. to long) längst (yet; for the longest time). Especially lang (long) may be "lank" in the extreme case.
Tang, Tank shows a clear difference, as do Zange (pliers) and zanke (1. p. sg. to twist, brawl, complain) except in noisy environments, but zankt (3. p. sg.) and *zangt (made up verbalisation of Zange, instead of zwingen (to force, obligate) zwicken (pinch) or whatever) not so clearly. The point is that ng can be stretched, but k not really.
nagt (he gnaws) and nakt (naked) show lengthening of the vowel and subvocal. Nacken (neck) nagen (to gnaw) are fully separated, but I have no clue how to spell nackend [or nakend?] (to be naked, as if participle of a nonexistend verb?). Nagel (nail) and Makel (flaw, blemish) where Makel gave Macke, because Makel invariably sounds shorter than Makel unless specifically stressed as if *Maakel. However, as pointed out in the comments below, Nagel has aspirated (voiced) g.
Genick (neck) and genug (enough) show barely any difference in termination. Neither do gesund (sound and healthy) and Stralsunt (that harbor near Denmark).
Mus (mousse) and Fuß (foot) are specifically differentiated in writing but not audibly so, to me; Muse (muse) /z/ is voiced, Muse (mousses) /s/ isn't. Likewise, Rus (but Russe) Bus (bus, as if *Buss, also Busse) aren't.
Knabbern is colloquial and might, sometimes, as well be transcribed knappern, I suppose (to gnaw, ie. on fingernails), cp. knapp (short), Knabe , Knappe (boy, cnave; difficult etymology).
Topf (pot) may be Topp, short o; toben (to play wild, to storm) 1. p. sg. tobe, tob', long o. That shows the fortition accross the whole word, where a voiced ablauting Toppe became fricative (if I say so myself), cp. Töpfer (potter). Schuppen (shed, shop) may be compared to Schub-lade, Schub (drawer), the later apparently from schieben (push, not pull nor draw) Schub (boost), schubsen (to shove, nudge), but both are compartments so maybe there's a closer relation.
Knabbern is colloquial and might as well be transcribed knappern, I suppose (to gnaw, ie. on fingernails), cp. knapp (short), Knabe , Knappe (boy, cnave; difficult etymology).
Staub (dust) and stopp (stop!) Stab (staff); cp. stopfen (to jam, plug) Stubbel (stub) Stoppel (bristle) Stumpf (stub, trunk) stumpf (blunt) Stube (living room, stove) steif (stiff) Stippe, stippen (dunk, touch) Stepp-Decke (woven blanket) Step-Tanz (step dance). So you can compare your own native stab step stop strobe (if you are native English speaking). Try saying stab-stab-stab quickly like a traicherous villain and stop-stop-stop in return (to reduce the lengthening of the a).
Dieb (thief) Dip, dippen (dip, related to stippen) Tipp, tippen (tip, hint; to tap). You will hardly find minimal pairs because they tended to dissimilate, e.g. tief (deep), long ago.