I want to rotate an object towards mouse position on screen in 3d environment. How can I do this?

Here's a fairly short and (IMO) sleek C# class file (a MonoBehaviour component) you can use for this kind of problem.

 using UnityEngine;

 public class MoveToMouse : MonoBehaviour
 {
     //  e.g. 0.2 = slow, 0.8 = fast, 2 = very fast, Infinity = instant
     [Tooltip("If rotationSpeed == 0.5, then it takes 2 seconds to spin 180 degrees")]
     [SerializeField] [Range(0,10)] float rotationSpeed = 0.5f;
 
   
     [Tooltip("If isInstant == true, then rotationalSpeed == Infinity")]
     [SerializeField] bool isInstant = false;
   
     Camera _camera = null;  // cached because Camera.main is slow, so we only call it once.

     void Start()
     {
         _camera = Camera.main;
     }
   
  
     void Update()
     {
         Ray ray = _camera.ScreenPointToRay(Input.mousePosition);
         Quaternion targetRotation = Quaternion.LookRotation(ray.direction);
    
         if (isInstant)
         {
             transform.rotation = targetRotation;
         }
         else
         {
             Quaternion currentRotation = transform.rotation;
   
             float angularDifference = Quaternion.Angle(currentRotation, targetRotation);  
             // will always be positive (or zero)
   
             if (angularDifference > 0) transform.rotation = Quaternion.Slerp(
                                          currentRotation,
                                          targetRotation,
                                          (rotationSpeed * 180 * Time.deltaTime) / angularDifference
                                     );
             else transform.rotation = targetRotation;
         }
     }
 }

Seems to work very well for me and seems to answer the question, so anyone who comes here can use/adapt/hack my code :) Let me know if I made any mistakes or you have some suggestions to improve, although it works super smoothly and intuitively in my star-fox style rail shooter game I'm making with Ben Tristem's GameDev_dot_TV course on Udemy ("C# Unity Developer 3D").

Here is a simple solution. It works but has it downsides.

     Camera cam;
 
     // Use this for initialization
     void Start () {
         cam = Camera.main;
     }
     
     // Update is called once per frame
     void Update () {
         Ray ray = cam.ScreenPointToRay(Input.mousePosition);;
         RaycastHit hit;
 
         if (Physics.Raycast(ray, out hit)) {
 
             Vector3 target = hit.point;
             target.y = 0;
             target.y = transform.localScale.y / 2f;
 
             transform.LookAt (target);
         }
     }

 using UnityEngine;
 public class Example : MonoBehaviour
 {
     //assign this field in inspector:
     public Transform subject;
     public float _depthOffset = 0f;
 
     Camera _camera;
 
     void Start ()
     {
         //copy a reference to main camera, for both convenience and performance:
         _camera = Camera.main;
     }
 
     void Update ()
     {
         //update rotation:
         Vector3 mousePositionWorldSpace = _camera.ScreenToWorldPoint( Input.mousePosition );
         Vector3 pointOfInterest = mousePositionWorldSpace+_camera.transform.forward*_depthOffset;
         Vector3 direction = pointOfInterest-subject.position;
         subject.rotation = Quaternion.LookRotation( direction );
     }
 }

I coded it so that the rotating object looks at the mouse even with or without a raycast hit. This should work. I basically combined the answers from @GregoryFenn and @Cornelis-de-Jager

 public Camera cam;
 public Transform obj;
 public float rotationSpeed;

 void Update()
 {
     Quaternion currentRotation = obj.rotation;
     Quaternion targetRotation;

     Ray ray = cam.ScreenPointToRay(Input.mousePosition);
     RaycastHit hit;
 
     if (Physics.Raycast(ray, out hit))
     {
         targetRotation = Quaternion.LookRotation(hit.point - obj.position);
     }
     else
     {
         targetRotation = Quaternion.LookRotation(ray.direction);
     }
     float angularDifference = Quaternion.Angle(currentRotation, targetRotation);

     if (angularDifference > 0)
     {
         obj.rotation = Quaternion.Slerp
         (
             obj.rotation,
             targetRotation,
             rotationSpeed * Time.deltaTime
         );
     }
     else
     {
         obj.rotation = targetRotation;
     }
 }