Is there a clear boundary between states of matter? Many physical properties of different substances, like melting and boiling points, are known already. They have discrete values at standard pressure (e.g., the boiling point of helium is at 4 K). The temperature 273.15 K, for example, corresponds to the amount of kinetic energy water molecules possess that breaks the intermolecular forces, notably hydrogen bonds.
However, by definition, at 273.15 K, H2O is a mixture of water and ice, which I find quite ambiguous. It is unclear at the very melting/boiling points of substances what state(s) they are in. As with H2O at 273.15 K, will H2O be liquid, a mixture of solid and liquid or entirely solid?
To what extent are intermolecular forces reduced at standard melting/boiling points?
 A: 
The temperature 273.15K, for example, corresponds to the amount of kinetic energy water molecules possess that breaks the intermolecular forces, notably hydrogen bonds.

First, a caution that temperature is not kinetic energy; this is an oversimplification (search temperature "kinetic energy" for many, many related discussion on this site). Moreover, ice sublimates into vapor at any temperature, so temperature doesn't provide a threshold for any particular bond being broken.

It is unclear at the very melting/boiling points of substances what state(s) they are in. As with H2O at 273.15K, will H2O be liquid, a mixture of solid and liquid or entirely solid?

The state of matter with the lowest Gibbs free energy $G$ is the equilibrium stable state. If $G$ is equal for two states at some temperature and pressure, then neither is more stable, and the fraction of each is indeterminate, as you note, without further information.

My question is, "To what extent are intermolecular forces reduced at standard melting/boiling points?"

This is something we can quantify, in a sense. The enthalpy of melting and vaporization of water is 333 J/g and 2257 J/g, respectively. Thus, we could say that as a whole or on average, water molecules become only 15% less well bonded—still in nearly the same close contact, but now with molecular rearrangement enabled—when they melt, with the remaining bonding lost upon vaporization. (Of course, we also have the distinctions that long-range order is lost upon melting and that surface tension ceases to be positive upon vaporization.) This arguably describes the phase change—an ensemble phenomenon—better than a discussion of individual forces between molecules.
This provides several ways to distinguish states of matter.
A: Symmetry breaking
Transitions from one state of matter to another state of matter are known as phase transitions or critical phenomena, and are accompanied by spontaneous breaking of a symmetry that exists in one state of matter, but not in the other (although there are some exceptions). The kind of symmetry broken may be more or less obvious: e.g., in case of crystallization a uniform and isotropic liquid or gaseous state is replaced (i.e., a state invariant to any translation or rotation) by a less symmetric crystal (which is invariant only to certain translations and certain rotations, allowed by the symmetry of the crystal lattice.)
One usually characterizes this situation by introducing an order parameter - a quantity that is zero in one phase, but not zero in the other. E.g., the average magnetization is characteristic of ferromagnetic phase transition - it is zero in the paramagnetic phase, and non-zero in the ferromagnetic one. In liquid-gas phase transition the order parameter is usually the difference between the liquid and gas density.
Phase coexistence
However, under certain conditions two phases of matter may coexist. The most common example here is the coexistence between liquid and gas beyond the critical point:

A: There is a clear boundary between a phase transition of matter, i.e. when and how matter will convert into another state. But there are no clear boundaries between matter states, i.e. in what form it should exactly be.
It's exactly the opposite,- at the triple point, namely at triple point temperature and pressure- $T_{tp}, P_{tp}$, matter will co-exist in all three phase states- gas, liquid, and solid. This is seen from a phase transition chart of water :

At $\approx 1~kPa$ pressure and $0~C^\circ$ temperature water co-exists as mixture of vapor,liquid and solid. Practically it looks like this (GIF quality was reduced to fit on the site):

A: 
However, by definition, at 273.15 K, H2O is a mixture of water and ice, which I find quite ambiguous. It is unclear at the very melting/boiling points of substances what state(s) they are in. As with H2O at 273.15 K, will H2O be liquid, a mixture of solid and liquid or entirely solid?

At 0°C and 1 atm ice melts, or water freezes. This means it is gradually, bit by bit, transformed from one phase to another.
At the melting/freezing conditions, substance in two phases coexists. This is sometimes described as mixed phase, but it is not some intermediary state between solid and liquid, it some solid and some liquid next to each other.
Melting requires adding heat, freezing requires evacuating heat. In mixed phase, the substance has enough heat to transform part of itself into liquid, while the rest of it is solid.
Therefore, the phase transition is a clear boundary.
As the others indicated, above the critical point, a substance can transform from liquid to gas without crossing any phase transition line. In such case, there is indeed no clear boundary.
A: My very layman's understanding is this:
Imagine you had a mass of ice below freezing, and gradually brought it up to exactly the boundary temperature between solid ice and liquid water (the melting/freezing point). If you stopped adding heat at exactly the boundary point, you would have entirely solid ice at that temperature. The ice does not instantly all transition to water as soon as it hits the boundary temperature (not even if you can manage to distribute the temperature perfectly evenly).
If you then add a little bit more heat, something different happens. The energy doesn't raise the temperature of the ice, instead it melts a little bit of it into water without heating it up. The ice actually needs energy to melt, beyond the energy going into changing its temperature. This is known as the enthalpy of fusion (or the enthalpy of vaporisation applies when transitioning from a liquid to a gas).
So in theory, you can have 100% ice that is at melting temperature. If you apply exactly the energy required for that amount of ice to transition to water, then you would have 100% water that is at the exact same temperature. If instead you only applied 50% of the energy required to melt that quantity of ice, then you would have a mixture of 50% ice and 50% water, still at the same temperature. (Of course in reality the temperature will not be perfectly even; you're much more likely to have ice at a range of temperatures from just below up to the melting point and water at a range of temperatures from the melting point to a bit above)
So there's a clear boundary in that each given "bit" of H2O is either water or ice, there's no state that is "in between ice and water" that is passed through on the way to melting the ice. The halfway point to melting a block of ice does not have all of the ice with "partially weakened bonds", or anything like that. But the transition of a large chunk of ice into water is a gradual process, not something that happens at a clear instant - more of the ice gradually undergoes the (sharp) transition to water as more energy is gradually added.
A: No. There is no clear boundary. There isnt even a boundary away from the transition point.
My understanding is that all materials exhibit nonzero vapor pressure under any reasonable circumstances.
Edit. The bot (and audience) apparently requires better articulation.
What youre envisioning as a "boundary" is better envisioned as an "equilibrium". The specific equilibrium relevant here is "vapor pressure". Read the wiki on vapor pressure, its quite good.
At any normal circumstance, all materials shed (i.e. "boil") atoms or molecules. For example, a cold bar of solid gold inside an experimental vacuum releases some nomzero number of atoms until an equilibrium is attained. This process exists at all temperatures, with atom shedding increasing with increasing temperature (and subsequently higher vapor pressure). That is, there is no physical boundary between the solid and its environment at any temperature not even away from a transition point. There is always exchange to pursue the equilibrium.
Furthermore, what youre attributing to "intermolecular forces" is really the realm of atmospheric pressure. Water boils because its vapor pressure exceeds the atmospheric pressure of its environment. kinectic energy of the individual atoms is changing ... sure, but thats somewhat of a red herring, as the balance is between states, and not dictated by whats happening "inside" the material. This is why phase diagrams are plotted against temperature and pressure. You cant conceptualize without firm attribution of state to both temp and pressure.
There is no physical boundary between states under any reasonable circumstances. No temperature or pressure state eliminates vapor pressure. There is always exchange. There is no material property disconuity at transition points or elsewhere. There is no adequately describing this process without equilibrium as a foundation.
Your question simply betrays a bias towards existing near STP. Some of these other answers miss the point entirely, or perhaps i dont understand the question myself.
