Court name abbreviations
(Table 7 in a nutshell)

Normally, the abbreviation for state intermediate appellate courts is "X Ct. App.," where X is the name of the state. But there are many exceptions, including the following:

Alabama Ala. Civ. App., Ala. Crim. App., Ala. Ct. App.
Connecticut Conn. App. Ct.
District of Columbia D.C.
Florida Fla. Dist. Ct. App.
Illinois Ill. App. Ct.
Maryland Md. Ct. Spec. App.
Massachusetts Mass. App. Ct.
New York N.Y. App. Div.
Oklahoma Okla. Crim. App.
Tennessee Tenn. Ct. App.; Tenn. Crim. App.
Texas Court of Criminal Appeals is Tex. Crim. App., (former Court of Appeals is Tex. Ct. App.)
Court of Appeals is Tex. App. (former Courts of Civil Appeals Tex. Civ. App.)

Court name abbreviations
(Table 7, full list)

The following alphabetical list provides abbreviations for court names to be used in cases according to rule 10.4.:

Administrative Court Admin. Ct.
Admiralty [Court, Division] Adm.
Alderman's Court Alder. Ct.
Appeals Court App. Ct.
Appellate Court App. Ct.
Appellate Department App. Dep't.
Appellate Division App. Div.
Armed Service Board of Contract Appeals A.S.B.C.A.
Bankruptcy Appellate Panel B.A.P.
Bankruptcy [Court, Judge] Bankr.
Board of Contract Appeals B.C.A.
Board of Immigration Appeals B.I.A.
Board of Tax Appeals B.T.A.
Borough Court [name] Bor. Ct.
Central District C.D.
Chancery [Court, Division] Ch.
Children's Court Child. Ct.
Circuit Court (old federal) C.C.
Circuit Court (state) Cir. Ct.
Circuit Court of Appeals (federal) Cir.
Circuit Court of Appeals (state) Cir. Ct. App.
City Court [name] City Ct.
Civil Appeals Civ App.
Civil Court of Record Civ. Ct. Rec.
Civil District Court Civ. Dist. Ct.
Claims Court Cl. Ct.
Commerce Court Comm. Ct.
Commission Comm'n
Common Pleas C.P. [when appropriate, name county or similar subdivision]
Commonwealth Court Commw. Ct.
Conciliation Court Concil. Ct.
County Court [name] County Ct.
County Judge's Court County J. Ct.
Court Ct.
Court of Appeal (English) C.A.
Court of Appeals (federal) Cir.
Court of Appeal[s] (state) Ct. App.
Court of Appeals for the Armed Forces C.A.A.F.
Court of Civil Appeals Civ. App.
Court of Claims Ct. Cl.
Court of Common Pleas Ct. Com. Pl
Court of Criminal Appeals Crim. App.
Court of Customs and Patent Appeals C.C.P.A.
Court of Customs Appeals Ct. Cust. App.
Court of Errors Ct. Err.
Court of Errors and Appeals Ct. Err. & App.
Court of Federal Claims Ct. Fed. Cl.
Court of [General, Special] Sessions Ct. [Gen., Spec.] Sess.
Court of Internationsl Trade Ct. Int'l Trade
Court of Military Appeals C.M.A.
Court of Military Review C.M.R.
Court of Special Appeals Ct. Spec. App.
Court of Veterans Appeals Ct. Vet. App.
Criminal Appeals Crim. App.
Criminal District Court Crim. Dist. Ct.
Customs Court Cust. Ct.
District Court (federal) D.
District Court (state) Dist. Ct.
District Court of Appeal[s] Dist. Ct. App.
Division Div.
Domestic Relations Court Dom. Rel. Ct.
Eastern District E.D.
Emergency Court of Appeals Emer. Ct. App.
Equity [Court, Division] Eq.
Family Court Fam. Ct.
High Court High Ct.
Judicial District Jud. Dist.
Judicial Division Jud. Div.
Judicial Panel on Multidistrict Litigation J.P.M.L.
Justice of the Peace's Court J.P. Ct.
Juvenile Court Juv. Ct.
Land Court Land Ct.
Law Court Law Ct.
Law Division Law Div.
Magistrate Division Magis. Div.
Magistrate's Court Magis. Ct.
Middle District M.D.
Municipal Court [name] Mun. Ct.
Northern District N.D.
Orphans' Court Orphans' Ct.
Parish Court [name] Parish Ct.
Police Justice's Court Police J. Ct.
Prerogative Court Prerog. Ct.
Probate Court Prob. Ct.
Public Utilities Commission P.U.C.
Real Estate Commission Real Est. Comm'n
Recorder's Court Rec's Ct.
Southern District S.D.
Special Court Regional Rail Reorganization Act Reg'l Rail Reorg. Ct.
Superior Court Super. Ct.
Supreme Court (federal) U.S.
Supreme Court (other) Sup. Ct.
Supreme Court, Appellate Division App. Div.
Supreme Court, Appellate Term App. Term
Supreme Court of Errors Sup. Ct. Err.
Supreme Judicial Court Sup. Jud. Ct.
Surrogate's Court Sur. Ct.
Tax Appeal Court Tax App. Ct.
Tax Court T.C.
Teen Court Teen Ct.
Temporary Emergency Court of Appeals Temp. Emer. Ct. App.
Territor[ial, y] Terr.
Traffic Court Traffic Ct.
Tribal Court [name] Tribal Ct.
Tribunal Trib.
Water Court Water Ct.
Western District W.D.
Workmen's Compensation Divison Workmen's Comp. Div.
Youth Court Youth Ct.