What are some algorithms which we use daily that has O(1), O(n log n) and O(log n) complexities?

If you want examples of Algorithms/Group of Statements with Time complexity as given in the question, here is a small list -

O(1) time

• Accessing Array Index (int a = ARR[5];)
• Inserting a node in Linked List
• Pushing and Poping on Stack
• Insertion and Removal from Queue
• Finding out the parent or left/right child of a node in a tree stored in Array
• Jumping to Next/Previous element in Doubly Linked List

O(n) time

In a nutshell, all Brute Force Algorithms, or Noob ones which require linearity, are based on O(n) time complexity

• Traversing an array
• Traversing a linked list
• Linear Search
• Deletion of a specific element in a Linked List (Not sorted)
• Comparing two strings
• Checking for Palindrome
• Counting/Bucket Sort and here too you can find a million more such examples....

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O(log n) time

• Binary Search
• Finding largest/smallest number in a binary search tree
• Certain Divide and Conquer Algorithms based on Linear functionality
• Calculating Fibonacci Numbers - Best Method The basic premise here is NOT using the complete data, and reducing the problem size with every iteration

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O(n log n) time

The factor of 'log n' is introduced by bringing into consideration Divide and Conquer. Some of these algorithms are the best optimized ones and used frequently.

• Merge Sort
• Heap Sort
• Quick Sort
• Certain Divide and Conquer Algorithms based on optimizing O(n^2) algorithms

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O(n^2) time

These ones are supposed to be the less efficient algorithms if their O(nlogn) counterparts are present. The general application may be Brute Force here.

• Bubble Sort
• Insertion Sort
• Selection Sort
• Traversing a simple 2D array

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O(n!) time

• Solving the travelling salesman problem via brute-force search
• generating all unrestricted permutations of a partially ordered set;
• finding the determinant with Laplace expansion
• enumerating all partitions of a set