indexOf Case Sensitive?

The indexOf() methods are all case-sensitive. You can make them (roughly, in a broken way, but working for plenty of cases) case-insensitive by converting your strings to upper/lower case beforehand:

s1 = s1.toLowerCase(Locale.US);
s2 = s2.toLowerCase(Locale.US);
s1.indexOf(s2);

There is an ignore case method in StringUtils class of Apache Commons Lang library

indexOfIgnoreCase(CharSequence str, CharSequence searchStr)

Yes, indexOf is case sensitive.

The best way to do case insensivity I have found is:

String original;
int idx = original.toLowerCase().indexOf(someStr.toLowerCase());

That will do a case insensitive indexOf().

Is the indexOf(String) method case sensitive?

Yes, it is case sensitive:

@Test
public void indexOfIsCaseSensitive() {
    assertTrue("Hello World!".indexOf("Hello") != -1);
    assertTrue("Hello World!".indexOf("hello") == -1);
}

If so, is there a case insensitive version of it?

No, there isn't. You can convert both strings to lower case before calling indexOf:

@Test
public void caseInsensitiveIndexOf() {
    assertTrue("Hello World!".toLowerCase().indexOf("Hello".toLowerCase()) != -1);
    assertTrue("Hello World!".toLowerCase().indexOf("hello".toLowerCase()) != -1);
}