Is there a way to find out which runway is used by which flight on a particular day/date?
(Guessing the) Runway Selection

For spotting airplanes, you want to know which are the current active runways. The runway history of planes or flights shouldn't matter much, unless your plane of interest is limited in what runways it can use.

Large airports have many parallel runways. In general, outsiders will not have enough information at hand to predict the active runways. The factors that are considered for the selection of active runways are the following:


  Aircraft take-off and land into the wind, or with minimal tail wind.
  Based on the wind direction, air traffic control will decide which
  runway is to be used at any given time.
  
  Larger airports tend to have more than one runway, so that a runway is
  always available depending on the wind direction. Airports with just
  one runway are generally constructed so that the runway is aligned
  with the prevailing wind.
  
  Runway selection is monitored at all times, as weather conditions can
  quickly change. Every runway has a wind indicator known as an
  ‘anemometer’, and wind observations contribute to the runway selection
  decision. Pilot reports of upper level winds can also impact on runway
  selection.
  
  When a runway is selected, it needs to be available for an extended
  period of time to allow pilots to plan their descent, approach and
  landing. As this involves anticipation of developing trends, aircraft
  may continue to land on a runway for a period when weather conditions
  at a local level no longer appear to warrant it.
  
  Other factors that air traffic control will take into consideration
  when deciding which runway to use include:
  
  
  the number and type of aircraft programmed for the airport
  length of runway(s)
  weather conditions (both present and forecast); including wind velocity and gradient, wind shear, wake turbulence effects and
  position of the sun
  availability of approach aids in poor visibility conditions
  location of other aircraft
  taxiing distances, including availability of taxiways
  braking conditions.
  
  
  Some airports also have ‘preferred runway’ systems. This means that if
  wind conditions, workload and traffic conditions permit, a particular
  runway will be used to move traffic as efficiently as possible to
  reduce the noise impact over residential areas.


This text is quoted from a short informational document on runway selection published by Airservices Australia.

ATC radio communication

It is possible to listen to ATC radio communications with a handheld scanner (always check if this is legal in your country!), or via the internet (LiveATC). There are different channels including Departure, Approach, Ground, and ATIS.

ATIS

Busy airports are usually equipped with Automatic Terminal Information Service. ATIS continuously broadcasts relevant information about the airport and immediate surroundings for pilots. This includes the actual runway in use. One can receive ATIS via VHF, or via phone. Frequencies and phone numbers are published in different ways. Airnav.com might be useful for the USA. LiveATC provides streaming of ATIS.

Ground

Aircraft are cleared for taxi to a runway on the Ground channel. 

Approach

Aircraft receive their instructions and clearance for a runway on the Approach channel.

Websites

Airports

Some airports are publishing their actual active runways on a website. For instance:


Amsterdam (Schiphol): "Baangebruik Schiphol".




This airport always assigns 1 active runway for landing and 1 active runway for starting. In the rush times, it opens an additional runway for landing or starting. In the snapshot above, the 2 runways for landing are red, the single runway for starting is blue.


Brussels: Runways in use


There are probably not that many airports publishing this information to non-pilots, but surely there are more. Please comment if you know any other!

Flight Tracking

If there is no information available from the airport directly, you can always use flight tracking web sites like the ones you have listed in your question to observe which are the current active runways, for starting and for landing respectively. Keeping in mind that the assignment of active runways is not random, it is always reasonable to guess that the next aircraft will use the same runway as the previous ones. For approaching aircraft it is often possible to see if it is directly pointing to a runway, or if it follows a pattern to a runway that other aircraft have taken before. 

Standard patterns and Operational procedures

If interested and willing to dig a bit deeper, you can always find and study standard approach patterns and operational procedures of your airport, e.g. 


Operational Procedures at Tokyo International Airport (Haneda)
search the internet for "RJTT SID STAR"


In combination with flight tracking web sites, this will allow you to recognize the usual patterns, and predict earlier where a tracked plane is going to land.