How can we tell if the Earth is spinning without any external references? The rotation of the Earth about its axis makes it bulge at the equator and contract at the poles due to the centrifugal forces. How do we know, without any external references, that the Earth is spinning if there is nothing to compare it to? For example, imagine it was spinning in empty space with no other objects. Does it still bulge at the equator? 
Particularly, what can we say theoretically about it?
 A: Other answers have correctly identified some practical observations that can be used to determine that Earth is rotating (gyroscope; tension in the rocks; Foucault pendulum; just dropping everyday objects, etc.). Here I'll add a remark about the background to the question. The background is an intuition that maybe general relativity suggests it would not be possible to tell that an object is spinning without looking outside. The main thing I want to say is that general relativity does not suggest that, except in a certain specific sense.
To get clarity about this, we need to distinguish two uses of the word 'rotating':


*

*something revolves relative to the distant stars

*a body is in rigid motion about an axis in a non-inertial manner


According to GR one can detect motion (2) without requiring observations of the rest of the universe. Just perform any of the experiments listed in other answers (and mentioned briefly in my opening paragraph). Furthermore, we can make a statement about what would happen if there were no "the rest of the universe". According to GR the rest of the universe does influence the local spacetime at any one point, but if one somehow removed the rest of the universe, and thus its influence on spacetime, then the local spacetime would not change very much, and one could still distinguish non-inertial from inertial frames by simple experimental tests. So one could still determine whether or not Earth had the motion of type (2) listed above.
Mach's principle was mentioned in another answer. The important point here is that the universe is not Machian in the most natural sense of the word (definitions in this area are not completely settled). In this sense Mach's principle is false.
Now a brief comment on type (1). It is possible to have an inertial frame (i.e. no "rotation" according to type (2)) which does revolve relative to the distant stars. This situation is called frame dragging; it occurs near to large rapidly rotating bodies. It is a tiny effect (i.e. such rotation is extremely slow unless something like a black hole is involved).
A: Earth rotation can be detected using a gyroscope (http://www.tkt.cs.tut.fi/research/nappo_files/Symposium_Gyro_Technology_2010_web.pdf). It looks like the equipment used in the article costs under $1000 (not including a computer).
EDIT (8/24/2019): Another approach was proposed by Compton (Science  23 May 1913:
Vol. 37, Issue 960, pp. 803-806):
"if a circular tube filled with water is placed in a plane perpendicular to the earth's axis, the upper part of the tube with the water in it is moving toward the east with respect to the lower part. If the tube is quickly rotated through 180 degrees about its east and west diameter as an axis, the part of the tube which was on the upper side attains a relatively westward motion as it is turned downwards (since it is drawing nearer the earth's axis). But the water in this part of the tube retains a large part of its original eastward motion, and this can be detected by suitable means."
"If then $\alpha$ is the angular velocity of the earth's rotation, $r$ the radius of the circle into which the tube is bent..., the relative velocity between the water and the tube when it is quickly turned from a position perpendicular to the earth's axis through 180 degrees is...$\alpha r$."
"In order to prevent convection currents, it is best to hold the ring normally in a horizontal position, in which case the relative motion is of course $\alpha r \sin\phi$, where $\phi$ is the latitude of the experimenter."
For $r=0.993 m$ and the latitude $40^{\circ} 48'33''$ (Wooster, Ohio), we obtain the velocity of about $0.05mm/s$; the average velocity observed by Compton agreed with the calculated velocity with 5% accuracy. Compton measured the water velocity by observing globules of oil introduced in the water using a micrometer microscope. 
A: Because the Earth is rotating on it-self, it is not an inertial referential, which means that there are additional fictitious forces acting on objects at rest in the frame of reference. For a spinning referential, the fictitious force is called the Coriolis force, which is responsable of many phenomena such as Foucault pendulum.
A: Using what NASA knows you could launch a small missile eastward from the equator and then launch an identical missile westward from the equator. If no spinning, the missiles go the same distance.  While this will work, the answer from The Photon seems more practical. lol
A: This is a Major debating point
A lot has been said theoretically about this. You have re-discovered an long running argument in the philosophy of physics.
It is often called the "bucket argument" as one of the most famous examples is a bucket of water that is spinning, and so the water level is not flat (bulges up the sides centrifugally) just like your example with the earth's seas bulging at the equator.
https://en.wikipedia.org/wiki/Bucket_argument
It is a longstanding problem amongst philosophers of physics. One of them (Ernst Mach) was especially concerned about this and his concerns fed into the development of general relativity. Mach believed that all motion must be relative, so that the centrifugal force driving the water to bulge outwards at the equator is actually caused by "the fixed stars" (somehow), so that in an empty universe with just the Earth alone you wouldn't know if it was spinning or not and the water would not bulge.
However, General Relativity failed to fix this problem. With GR in an empty universe with just a bucket of water (or the Earth), the bucket/Earth still knows if it is spinning or not. 
Their are two camps on what this means:
(1) - Our best theory of physics tells us that rotation is absolute not relative. So it probably is. Stuff "just knows" when it is spinning, this needs no special explanation.
(2) - This is a really big problem, and shows that General Relativity isn't entirely right. One day we will find a better theory that fixes it so that things can only tell when they are spinning through interaction with other things.
Their are people in both camps. We may never know which group is right.
A: Earth's gravity is countered by the centrifugal force to a measurable degree. This is one of the reasons the free-fall acceleration is smaller at the equator, compared to the poles.
