What is the origin of the phrases "up the creek" and "up shit's creek," and which is the original version?

Refer to the history of the Republican river. Pawnee Indians referred to it as "Manure River" in the late 1700's and early 1800's because of the large number of Buffalo in this area and the river's inadequate size to "flush" itself.

Up the proverbial estuary with no mechanical means of propulsion.

The origin answers here lack dating of the phrase and variations, along with a lack of citations, which renders them totally useless.

This saying originates from Gosport England Hasler to be exact its from where injured saliors would be sent up the creek, a river in Hasler and then loaded onto another boat that was on rails and then pulled up to Hasler Hospital (Parts of the rail system are still there) hens the saying UP THE CREEK WITHOUT A PADDLE.

this dates back to the 18th century and originated in london. If you did not pull your weight as a sailor or deck hand, on return to the port of London the lazy sailor would be cast off in a small boat with no oars or paddles into bow creek. At this time was an open sewer full of shit and animal carcases. The poor fellow would the have to use his hands to paddle the boat through the shit to reach safety. Hence up shit creek.

The more powerful is the second phrase but they originated at the same time. The word "shit" by the way is a very old word that was used in 13th century England. The verb to shit dates the Middle English period and the noun form is from the 16th century. The interjection is of quite recent vintage, not found until the 1920s.

“Up the creek”
Fail this test and you'll be swimming up the creek.

"Up shit creek without a paddle"
To be in a bad situation which is either slowly or rapidly getting worse with no evident way out. Should you be armed with a paddle, you still, typically, have a slow long recovery. The phrase is a gentrified version of a WWII saying "Up shit's creek without a paddle" which summons up a mental picture more graphic than the current refined one... Imagine being in a rowboat at the top of Niagara Falls and you lose an oar! His first parachute wouldn't open, then the emergency one failed - you might say he was up the creek without a paddle.

“Up shit creek”
To be in an undesirable place, or to be in trouble. A contraction of "Up shit creek without a paddle." It can be made to define even more extreme circumstances by adding the suffix "with the repo men on the shore waiting for the boat"

from the net.

If you are up a creek without a paddle, that isn't a problem...since once you are already up the creek, you are there. Now, if you need to go down the creek, that isn't a problem since the current will take you there. What is really a problem though it to NEED to go up the creek when you do not have a paddle (or a motor). Therefore, the expression is better suited to say "that is a heluva mess to need to go up the creek without a paddle."