Here are examples for the most common lifecycles:
componentDidMount
Pass an empty array as the second argument to useEffect() to run only the callback on mount only.
function Example() {
  const [count, setCount] = useState(0);
  useEffect(() => {
    document.title = `You clicked ${count} times`;
  }, []); // Pass an empty array to run only callback on mount only.
  return (
    <div>
      <p>You clicked {count} times</p>
      <button onClick={() => setCount(count + 1)}>
        Click me
      </button>
    </div>
  );
}
componentDidUpdate (loose)
By passing just the single argument into useEffect, it will run after every render. This is a loose equivalent because there's a slight difference here being componentDidUpdate will not run after the first render but this hooks version runs after every render.
function Example() {
  const [count, setCount] = useState(0);
  useEffect(() => {
    document.title = `You clicked ${count} times`;
  }); // No second argument, so run after every render.
  return (
    <div>
      <p>You clicked {count} times</p>
      <button onClick={() => setCount(count + 1)}>
        Click me
      </button>
    </div>
  );
}
componentDidUpdate (strict)
The difference of this example with the example above is that the callback here would not run on initial render, strictly emulating the semantics of componentDidUpdate. This answer is by Tholle, all credit goes to him.
function Example() {
  const [count, setCount] = useState(0);
  const firstUpdate = useRef(true);
  useLayoutEffect(() => {
    if (firstUpdate.current) {
      firstUpdate.current = false;
      return;
    }
    console.log('componentDidUpdate');
  });
  return (
    <div>
      <p>componentDidUpdate: {count} times</p>
      <button
        onClick={() => {
          setCount(count + 1);
        }}
      >
        Click Me
      </button>
    </div>
  );
}
componentWillUnmount
Return a callback in useEffect's callback argument and it will be called before unmounting.
function Example() {
  const [count, setCount] = useState(0);
  useEffect(() => {
    // Return a callback in useEffect and it will be called before unmounting.
    return () => {
      console.log('componentWillUnmount!');
    };
  }, []);
  return (
    <div>
      <p>You clicked {count} times</p>
      <button onClick={() => setCount(count + 1)}>
        Click me
      </button>
    </div>
  );
}
shouldComponentUpdate
You can already achieve this on the component level using React.PureComponent or React.memo. For preventing rerendering of the child components, this example is taken from React docs:
function Parent({ a, b }) {
  // Only re-rendered if `a` changes:
  const child1 = useMemo(() => <Child1 a={a} />, [a]);
  // Only re-rendered if `b` changes:
  const child2 = useMemo(() => <Child2 b={b} />, [b]);
  return (
    <>
      {child1}
      {child2}
    </>
  )
}
getDerivedStateFromProps
Again, taken from the React docs
function ScrollView({row}) {
  let [isScrollingDown, setIsScrollingDown] = useState(false);
  let [prevRow, setPrevRow] = useState(null);
  if (row !== prevRow) {
    // Row changed since last render. Update isScrollingDown.
    setIsScrollingDown(prevRow !== null && row > prevRow);
    setPrevRow(row);
  }
  return `Scrolling down: ${isScrollingDown}`;
}
getSnapshotBeforeUpdate
No equivalent for hooks yet.
componentDidCatch
No equivalent for hooks yet.