Pi is, if nothing else, the universe's perfect random number generator. Every counting number (1, 2, 3, 4...) occurs with equal consistency throughout samplings of the transcendental number, and in a completely random fashion. Thus, much like my room with 1,000 typing monkeys, if you let pi unspool long enough, you'll find anything. My name. An MP3 of that damn Cibo Matto song I can't seem to get my hands on. Anything.

Hence, there are no "patterns" in pi. People who look for them ignore the infinite possibilities of infinite strings of random numbers.

But, for starters:
3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744

Note: The assertion that the digits of pi are normal is widely conjectured but unproven. (Also note that random is not the same thing as normal. Hate to hairsplit, but the digits of pi are certainly not random). It is important to note that the digits of certain other irrational numbers appear to be normal to a superficial inspection but in fact are not (such as root three, for instance. at least, so I've heard.)

An actual proof that the digits of pi are in fact normal would make big waves in the math circles.