Frank Zappa is cultish I suppose . I highly recommend either Apostrophe or Overnight Sensation . Frank was in The Mothers of Invention with Captain Beefheart by the way .
If by cult we mean dedicated fan following but slightly out of the mainstream , then Fairport Convention are brilliant . Leige and Leith is considered the classic album . Whilst outside of the rock genre , Waylon Jennings has a hugely dedicated following .
Parliament / Funkadelic if you feel like getting funky . Dusty Springfield if you feel like getting soulful .
I'm not really your "go to guy" for more modern stuff though . Most of the acts I have mentioned are reasonably well known , however they have a following of obsessive fans that far outweighs their level of mainstream success
EDIT.....this actually really got me thinking , especially if the criteria is that you are not solely after weird music , but quality too . So many genres have their cult faves .
Jerry Lee Lewis , a monster of 50's rock n roll but often overshadowed . Thelonius Monk if you want some really quirky jazz . Jefferson Airplane are , for me , the quintessential psychedelic 60's group . The Kinks are unique amongst the British Invasion groups . Mickey Newbury is one of the all time great American singer songwriters and sadly unappreciated .
Once again I stress many of these artists I have not selected for the quirkiness of their music unlike Beefheart of Zappa , but for the esteem they are held in by their fans
