Should I register pod or kubernete service to consul on kubernetes cluster

• Headless Service is the answer

Kubernetes environment is more dynamic in nature.

deregister a service when the pod is dead

Yes

Kubernetes Pods are mortal. They are born and when they die, they are not resurrected. While each Pod gets its own IP address, even those IP addresses cannot be relied upon to be stable over time. A Kubernetes Service is an abstraction which defines a logical set of Pods and provides stable ip

That's why it is recomended to use headless service which basically fits into this situation. As they mentioned in first line in docs

Sometimes you don’t need or want load-balancing and a single service IP. In this case, you can create “headless” services by specifying "None" for the cluster IP (.spec.clusterIP)

headless service doesn't get the ClusterIP. If you do nslookup on the headless servive, it will resolve all IPs of pods that are under headless service. K8s will take care of adding/managing pod IP under the headless service. Please for more details. And I believe, you can register/provide this headless service name in Cosule.

• Please refer this blog for detailed here

UPDATE1:

Please refer this Youtube video. May give you some idea.(Even I have to watch it..!!)