IFR Alternate Required
The 1-2-3 Rule for determining if an alternate is required comes from CFR §91.169 as highlighted below. The way the FAR is written an alternate is always required unless certain conditions are met. Basically, if your destination has an instrument approach then, for 1 hour before to 1 hour after your anticipated arrival at your destination the weather does not have a ceiling of 2,000′ and visibility of 3 miles, an alternate is required. The alternate must have a ceiling of 600′ and visibility of 2 statute miles for precision approaches and 800′ and 2 miles for non-precision approaches unless non-standard alternates are published for the approach. If there is no published approach at the alternate, then the ceiling and visibility minima are those allowing descent from the MEA, approach, and landing under basic VFR.
Scenarios:
Destination Palm Springs (KPSP). The airport has multiple approaches and weather reporting. But unless we are RNP equipped, we can only fly the VOR or GPS-B approach. Apply the 1-2-3 rule to see if an alternate is required. However, note that the minimum ceiling for the approach is 2,300′ so even if you meet the 1-2-3 rule, you may still not be able to land. An alternate would be a good idea in that case.
Destination Bermuda Dunes (KUDD). The airport has multiple approaches but no weather reporting. The nearest weather forecast (TAF) is 8 miles away at Palm Springs (KTRM) and 10 miles away at Palm Springs (KPSP). There is no way to apply the 1-2-3 rule so an alternate is required.
Destination Borrego Valley (L08). There is weather reporting but no forecast and one GPS approach. An alternate is required.
Destination King City (KKIC). There is weather reporting but no forecast and no approaches. An alternate is required.
Destination Harris Ranch (308). There is no weather reporting, no forecast, and no approaches. An alternate is required.
Alternate Bermuda Dunes (KUDD). The airport has multiple approaches but no weather reporting but all three of the approaches show 
in the Pilot Briefing section of the approach plate so they are not available as an alternate. This is true even if, by looking at the weather at KTRM and KPSP you can determine that the ceiling and visibility in the area would allow descent from the MEA, approach, and landing under basic VFR.
Alternate Palm Springs, Jacqueline Cochran International (KRTM). The airport has two GPS approaches and two VOR approaches. All of them show 
in the briefing section, so we can check to see what the restrictions are and see that it just bumps the ceiling and visibility from standard. If the weather at the time of arrival meets those minimums, then we can file it as an alternate. When we get there we can use any approach that the weather permits.
Alternate Borrego Valley (L08). There is one approach showing in the Pilot Briefing section of the approach plate. It is not available as an alternate because when you look up the alternate minimums, it says “NA when local weather not available” and there is no TAF.
The rule for filing with GPS is not spelled out in the FAR. This post goes into detail. Basically, if you have a non-WAAS GPS you can file either the destination or alternate with a GPS approach—but not both. If you have WAAS, you both can have only GPS IAPs. You must to use the LNAV or circling minimums at the destination.
§91.169 IFR flight plan: Information required.
(a) Information required. Unless otherwise authorized by ATC, each person filing an IFR flight plan must include in it the following information:
(1) Information required under §91.153 (a) of this part;
(2) Except as provided in paragraph (b) of this section, an alternate airport.
(b) Paragraph (a)(2) of this section does not apply if :
(1) Part 97 of this chapter prescribes a standard instrument approach procedure to, or a special instrument approach procedure has been issued by the Administrator to the operator for, the first airport of intended landing; and
(2) Appropriate weather reports or weather forecasts, or a combination of them, indicate the following:
(i) For aircraft other than helicopters. For at least 1 hour before and for 1 hour after the estimated time of arrival, the ceiling will be at least 2,000 feet above the airport elevation and the visibility will be at least 3 statute miles.
(ii) For helicopters.…
(c) IFR alternate airport weather minima. Unless otherwise authorized by the Administrator, no person may include an alternate airport in an IFR flight plan unless appropriate weather reports or weather forecasts, or a combination of them, indicate that, at the estimated time of arrival at the alternate airport, the ceiling and visibility at that airport will be at or above the following weather minima:
(1) If an instrument approach procedure has been published in part 97 of this chapter, or a special instrument approach procedure has been issued by the Administrator to the operator, for that airport, the following minima:
(i) For aircraft other than helicopters: The alternate airport minima specified in that procedure, or if none are specified the following standard approach minima:
(A) For a precision approach procedure. Ceiling 600 feet and visibility 2 statute miles.
(B) For a nonprecision approach procedure. Ceiling 800 feet and visibility 2 statute miles.
(ii) For helicopters: …
(2) If no instrument approach procedure has been published in part 97 of this chapter and no special instrument approach procedure has been issued by the Administrator to the operator, for the alternate airport, the ceiling and visibility minima are those allowing descent from the MEA, approach, and landing under basic VFR.
(d) Cancellation. When a flight plan has been activated, the pilot in command, upon canceling or completing the flight under the flight plan, shall notify an FAA Flight Service Station or ATC facility.