I have a Dell Inspiron 5558 and my laptop's fan would only spin for a second during the time I pressed the power button and it would never spin again but my laptop would still boot and turn on. Except for one time after a few hours of pressing the power button when it just started to spin again and when I shut down my laptop and turned it on again, it would only spin at the start and would won't spin again
Could be a loose connection, could be the fan is going bad. I would suggest either contacting the manufacturer (if under warranty) or trying a local tech if no warranty. Unless you yourself feel comfortable opening it up and checking it. Then possibly replacing it.
You may even find a video or two that can walk you through it online.