B737 Undifferentiated or Other Model
Flight Crew Instrument; Flight Crew Air Transport Pilot (ATP); Flight Crew Multiengine
Flight Crew Last 90 Days 155.35; Flight Crew Type 2355.80
ATC Issue All Types; Inflight Event / Encounter Weather / Turbulence; Inflight Event / Encounter Fuel Issue
Air Traffic Control Issued New Clearance; Flight Crew Requested ATC Assistance / Clarification; Flight Crew Diverted
Weather
While descending on the RNAV [arrival] to ZZZ; ZZZ closed airport; [and we] entered holding. Burned into fuel; left us with 11.0 when released from hold.  Subsequently; vectored for approach ILS XXR.  Bingo Fuel was 10.00; Burn to Alternate (ZZZ1) 4.0.   ZZZ1 weather deteriorating.   ZZZ Tower reporting braking 3-3-3; 1/8 inch wet snow on top of ice. Aircrafts ahead of us reporting LLWS [low level wind shear]; resulting in them going around due to cockpit wind shear warning. Numerous wind shear reports and rapid pressure changes.  We decided to divert to ZZZ2; since ZZZ1 alternate was deteriorating. On way to ZZZ2; airport reported LLWS and microburst at field.  Asked for clearance to ZZZ3; declared MIN FUEL. After more research; found ZZZ4 to be a better choice both for weather and for fuel and asked for clearance to ZZZ4.  Asked for expedited handling with little success. [Finally] received priority handling and landed with 3;700 lbs of fuel; uneventful approach and landing.
B737 Captain reported requesting expedited handling to an alternate due to low fuel situation following encounter with significant weather at destination.