Inconsistent verdicts
Inconsistent verdicts can come up in various ways.
Acquittal on one count should mean acquittal on another count
California law is pretty clear. Penal Code section 954 says "[a]n acquittal of one or more counts shall not be deemed an acquittal of any other count." People v. York (1992) 11 Cal.App.4th 1506, 1510 held that PC954 means that jury can render inconsistent verdicts and that inconsistent verdicts are not grounds for motion for a new trial. And there's California Supreme Court cases as recent as this year maintaining that. (People v. Carbajal (2013) 56 Cal.4th 521, 532.) The court is supposed to respect the jury's verdict even if it can result in nonsensical situations such as in a rape-murder case, acquitting on the rape, but finding the rape-murder special circumstance true. (People v. Avila (2006) 38 Cal.4th 491, 600.) The courts have rationalized inconsistent verdicts as lenity, compromise, or mistake on the part of jurors, and that none of those is a reason for overturning a jury verdict. That a judge may review a verdict for sufficiency of the evidence is supposedly a safeguard for crazy jurors. Another safeguard is Penal Code section 1161, where the judge is supposed to make sure the jury did not make a mistake of law. The United States Supreme Court has said that inconsistent verdicts are okay, analogizing it as if each count was tried separately, and the exact same evidence was presented in each separate trial, the inconsistent verdicts across the separate trials would be okay, so it should follow that inconsistent verdicts in one trial is okay. (Dunn v. United States (1932) 284 U.S. 390, 393.) The Supreme Court has even said that there is no constitutional implications for inconsistent verdicts. (United States v. Powell (1984) 469 U.S. 57, 65.)
However, there is an ancient California case, People v. Walther (1938) 27 Cal.App.2d 583, which says an acquittal on one count can require the reversal of a conviction on another count.
Conviction on one count logically means acquittal on another count
United States v. Powell said "a situation where a defendant is convicted of two crimes, where a guilty verdict on one count logically excludes a finding of guilt on the other" might be different. (Powell, at p. 69 fn. 4.) The 9th Circuit has said that in this situation there might be grounds for a due process challenge. (Masoner v. Thurman (1993) 996 F.2d 1003, 1005.) There are some really old California cases which do say inconsistent jury verdicts must be overturned.
People v. Calpito (1970) 9 Cal.App.3d 212 said inconsistent jury verdicts can't be allowed if the elements of the different offenses are identical. People v. Tophia (1959) 167 Cal.App.2d 39 said that inconsistent verdicts won't stand when there is a "real factual inconsistency."
Lesser Included Offenses
Conviction on both greater and lesser offense bars conviction on the lesser. (People v. Ward (1986) 188 Cal.App.3d 459, 472.)