Should I use a Singleton to instantiate my service class?

I'm currently working on a .NET 4.7.1 application. We don't use Dependency Injection in our project.

I extracted some logic into a service class, which gets called on an HTTP request.

Currently I instantiate my service class like a regular private member.

I was just wondering if I should implement the Singleton pattern into my Service class to perhaps improve performance. Hence not to instantiate a new service instance on each HTTP request.

My code looks like this:

public class TheServer
{
   private static readonly object locker = new object();
   private TomatoService tomatoService => new TomatoService();

   private void SendVegies()
   {
      lock(locker)
      {
          var result = tomatoService.CreateSalad();

          // ... some send method call, that's why I need to lock the thread
      }
   }
}

public class TomatoService
{
   public List<string> CreateSalad()
   {
      // some logic
      // return someTomatoSalad;
   }
}

Everything seems to work fine, I am just concerned about my code quality. Should I write my TomatoService as a Singleton class?

Would a Singleton TomatoService be better for the performance?

What do you think?

Thank you very much!