B737 Captain reported that a slat failure required a diversion to a nearby airport after a consultation with dispatch and maintenance did not remedy the situation.

Narrative

During preflight the maintenance release showed a maintenance verification flight. The release requested an APU inflight start. The APU failed to start on the previous verification flight. We determined we would not be able to complete the verification because our flight time was scheduled to be XA:26. During boarding and about 25 minutes to scheduled departure; Maintenance arrived at the airplane and said they had been directed to defer the APU. We reviewed and briefed the engine air start and cross-bleed start checklist; and briefed face to face with ramp personnel. During taxi out and after the cross-bleed start; we got zone temp lights for the CONT CABIN and FWD CABIN. We taxied to the holding pad to troubleshoot the zone temp lights with Maintenance. The decision was made to defer the primary channels of the zone temp controllers. After an uneventful take-off; the flaps retracted normally. During arrival to ZZZ; we noticed the primary left flap indicator needle was bouncing. We decided to configure the flaps early to ensure the flaps would function normally. We selected flaps 1; but the slats failed to extend. We verified the flap indication on the primary flap indicator and the secondary flap indicator on the aft overhead panel; but we did not have a LE TRANSIT light or a flaps 1 indication on the airspeed tape. We were on downwind for Runway XX to ZZZ and asked ATC for delayed vectors to troubleshoot the problem. We did not have a caution light but we had no slat indication on both flap indicators. We started with the flap transit checklist. The first bullet point states; 'One or more of these may occur: the leading edge devices are not in the commanded position.' Therefore; we determined this was the appropriate checklist. Step 3 of the checklist directs you to select flaps up then reselect flaps 1. After completing step 3 we still had no slat indications on the primary; secondary flap indicators or the airspeed tape; but we also still did not have a caution light or an flap transit light. Fuel was a concern so we set a timer for 10 minutes to further troubleshoot. The checklist does not address a no caution light with no flap indications. We attempted to call Maintenance and got a frequency from Dispatch. Dispatch also said we were probably too low for reception. When we got on the frequency it was occupied with another aircraft. ATC asked us if we were requesting priority handling and we told them to standby. They then said the company was requesting priority handling for us for a flight control malfunction. We had every intention to request priority handling; but we were still engaging the checklist. The next checklist we ran was the all flaps up landing checklist. Step 1 states; 'Accomplish this checklist only by the flaps-trailing edge flaps up landing checklist.' We did not have a disagreement or asymmetry indication. We had no indications. We determined this checklist did not apply and step 4 sent us to the no flap landing checklist. After running the no flap landing checklist and checking the non-normal landing configuration tables; we elected to divert to ZZZ1 and land on Runway XXC. We briefed the Flight Attendants and told them no evacuation preparation was necessary. We had Airport Rescue and Firefighting (ARFF) standing by as a precaution. Approach speed was 196 kts. We used maximum manual braking and full reverse thrust and exited the runway. The landing was very smooth and braking was normal. After parking at the gate we debriefed each other of what went well and what could have gone better; then debriefed the Flight Attendants; Maintenance; and Dispatch.