There is hardly any soil property which does not affect availability of nutrient to plants . These soil properties can be broadly grouped into soil biological, physical , chemical, geological, petrological, mineralogical properties conditioned by the climatic parameters over space and time..Any classical work on Mineral Nutrition of plants will give sufficient idea..

Prof Anoop said it well. Nutrient availability to plant is affected by soil properties (e.g. soil acidity, OM content, etc), biotic factors such as plant root distribution, density, depth, microbes, etc and type of nutrient. These are general.

the following factors can affect micronutrient availability:

Soils low in organic matter (less than 2.0%)

Soils with higher amounts of clay

Soils that have very high levels of organic matter (greater than 30% organic matter to a depth of 30 cm)

Cool, wet soil conditions

Soils with pH levels higher or lower than 6.0 and 7.0

Thank you Dear Prof Anoop, Nihad and Iticha,  I would like to add tow parameters are important that influence of soil minerals on the solubility and availability of nutrients to plant, there are (intensity and capacity). intensity which is the concentration of an element to soil solution and capacity which is the ability of solid phases in soils to replenish that element as it is depleted from from solution 

These soil parameters , when you talk in terms of a specific crops, are highly soil condition specific. For example, high rate of tuberisation insanity loan soils compared to clay soils, tea/ coffee preceding acid soil condition than alkaline soil conditions, seed spices grow excellently in salt affected soils, like that you can cite many other examples. You can't consider soil parameters alone with nature of crop and it's requirement..

Soil pH,EC,OM content , soil structure,and texture,availability of the element in the soil for absorption by plant roots,the ionic state of the element , Soil temperature,microorganisms population ,all these traits give us a good idea about availability of an element to plant.

In addition to what are mentioned above by our friends, the form of the nutrient or element and its interaction with soil also affects availability. For instance, NO3- are vulnerable to leaching on vermiculite soils and performs good on kaolinite soils (soils dominated by positive charged surfaces). Conversely, NH4+ is more stable and available to plants on negatively charged colloids than positively charged colloids. This is related to nutrient retention in soils.  

Off course , then nutrient interaction , their ratios and consequently the nutrient ratios in index plant irgans , they all are going to matter in ultimate plant nutrition which is nothing but a translatioon of soil fertility only..vis-a-vis  soil properties..what we popularly call as nutrient use efficiency..

In addition for all respected information , I would like to display my experiency  by using biocontrol agent specialy prpmoting fungi like trichoderma spp ,chaetomium spp,michorhizal fungi ,they  play asignificant role in availability of elements to plants.

Plus presence of optimum environment for nitrogen fixing and phosphorus solublizing bacteria. In fact all life in the soil directly or indirectly affects nutrient supply and availability to the plant.

Interesting answers... to add to the list:  I think what we need to consider is the primary pools or reservoirs of nutrients in a soils which involve: Soil solution; Exchangeable cations and anions; Sorbed cations. on another side  we have to look at the synergistic and antagonistic effects of different nutrient.. thirdly timely and optimally available water and good soil water and  plant  relationship; of course the growth stage  and root anatomy of the plant are some of the things that impede or facilitate nutrient availability to plant..

Nutrient indexing vis-a-vis crop is another very missing issue in the whole discussion..