For state supreme courts a total of 38 states have some type of judicial elections. The breakdown of selection systems for state high courts is as follows:
Seven (7) states have partisan elections (AL, IL, LA, NC, PA, TX, WV; All judges in both Illinois and Pennsylvania run in uncontested retention elections for additional terms after winning a first term through a contested partisan election)
Fourteen (14) states have nonpartisan elections (AR, GA, ID, KY, MI, MN, MS, MT, NV, ND, OH, OR, WA, WI; Ohio and Michigan have nonpartisan general elections, but political parties are involved with the nomination of candidates, who frequently run with party endorsements)
Seventeen (17) states have uncontested retention elections after initial appointment (AK, AZ, CA, CO, FL, IN, IA, KS, MD, MO, NE, NM, OK, SD, TN, UT, WY; All judges in New Mexico are initially appointed, face a contested partisan election for a full term, and then run in uncontested retention elections for additional terms)
The remaining 12 states grant life tenure or use reappointment of some type for their highest courts (CT, DE, HI, MA, ME, NH, NJ, NY, RI, VT, VA, SC)
In Alabama (for example), which has partisan elections for Supreme Court Justices and retention elections, all the Justices are Republicans and they voted 7-2 that the Alabama Secretary of State did not have the authority to investigate Barack Obama’s eligibility.
I'm not sure you are stating that accurately, I think they stated that the Secretary of State could not be COMPELLED to do so. It appears that they can do so if they want to, but the public can't make them do their jobs.
We know that the California Election people kicked Roger Caldero off the ballot for being born in Nicaruagua, so it's obvious that they can if they want to. We just can't MAKE them do so when they don't want to.
Apparently they have the same sort of discretionary powers as do the Obama puppet "justice" department in refusing to prosecute Lois Lerner.
What used to be regarded as "noble privilege."