The simplest thing to do is:

Collections.sort(list, String.CASE_INSENSITIVE_ORDER);

kindly follow this way, don't make code too much stuff, here i see you have mess up you code and it makes difficult to understand easily.

kindly follow this easiest way to sort Dog Object's list in sorting order(Ascending/Descending) way.

import java.util.ArrayList;
import java.util.Collections;


class Dog implements Comparable<Dog>{
    String name;

    Dog(String name){
        this.name = name;
    }

    public String getName() {
        return name;
    }

    public void setName(String name) {
        this.name = name;
    }

    @Override
    public int compareTo(Dog o) {
        return this.name.compareTo(o.getName()); // dog name sort in ascending order 
        //return o.getName().compareTo(this.name); use this line for dog name sort in descending order 
    }

    @Override
    public String toString() {
        return this.name;
    }
}


public class DogSort {

    public static void main(String[] args) {
        ArrayList<Dog> listDog = new ArrayList<Dog>();
        listDog.add(new Dog("doggy"));
        listDog.add(new Dog("aaa"));
        listDog.add(new Dog("bbb"));
        System.out.println(listDog);
        Collections.sort(listDog);
        System.out.println(listDog);

    }

}

Although the question already has an accepted answer I would like to share some Java 8 solution

// if you only want to sort the list of Vehicles on their email address
Collections.sort(list, (p1, p2) -> p1.getEmail().compareTo(p2.getEmail()));

.

// sort the Vehicles in a Stream
list.stream().sorted((p1, p2) -> p1.getEmail().compareTo(p2.getEmail()));

.

// sort and print with a Stream in one go
list.stream().sorted((p1, p2) -> p1.getEmail().compareTo(p2.getEmail())).forEach(p -> System.out.printf("%s%n", p));

.

// sort with an Comparator (thanks @Philipp)
// for the list
Collections.sort(list, Comparator.comparing(Vehicle::getEmail));
// for the Stream
list.stream().sorted(Comparator.comparing(Vehicle::getEmail)).forEach(p -> System.out.printf("%s%n", p));

Sort an array with Arrays.sort(Hospitals).

As suggested in the comments, the most simple way of maintaining a sorted data structure upon each insert is to use a TreeSet or any other data structure that maintains sorted order internally. Instead of declaring an ArrayList<String> you would simply need to modify your code to this:

Set<String> names = new TreeSet<>();
Scanner scan = new Scanner(System.in);
String name;
do {
  System.out.println("Enter the next name: ");
  name = scan.nextLine();
  String toUpperCase = titleCase(name);
  if(!toUpperCase.equals("Stop")){
    names.add(toUpperCase);
  }
} while(!name.equalsIgnoreCase("STOP"));

From Javadocs for TreeSet:

the Set interface is defined in terms of the equals operation, but a TreeSet instance performs all element comparisons using its compareTo (or compare) method, so two elements that are deemed equal by this method are, from the standpoint of the set, equal. The behavior of a set is well-defined even if its ordering is inconsistent with equals; it just fails to obey the general contract of the Set interface.