Before you start looking for fossils on the streets you should learn to identify the rocks where fossils are usually found. You should also take into account the groups of rocks that cannot possibly contain fossils, so that you focus your search on construction materials where urban fossils are most likely to be found.

Where you should NOT look for fossils: igneous and metamorphic rocks
Rocks formed into the Earth’s crust, where temperature and pressure are very high, do not preserve fossil remains because either life does not exist in those conditions or the organic that might be preserved in the geological materials has been completely obliterated. There are two kinds or rocks where you will never find fossils: igneous and metamorphic rocks. On the one hand, Igneous rocks are formed when magma cools down and they are made of crystals of minerals, although when this process takes place more slowly, the rock is made of glass. On the other hand, metamorphic rocks are formed when preexisting rocks (either sedimentary, igneous or even metamorphic rocks) undergo extreme conditions of high pressure and temperature that modify their internal structure and the mineral composition. This process often destroys fossils in sedimentary rocks but when metamorphism is relatively low some fossil remains may remain.

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Examples of construction stones without fossils: [A] polished igneous rock, [B] unpolished igneous rock, [C] wall made of slate, a metamorphic rock and [D] marble, another kind of metamorphic rock.

Where you SHOULD look for fossils: sedimentary rocks

Fossils are found on sedimentary rocks, that is to say, sediment that has undergone a lithification process. Burying gives way to a number of changes in this sediment, such as dewatering, cryistalization of minerals, etc., which eventually transforms the sediment into a rock. Some sedimentary rocks are made of solid particles of various sizes, such as conglomerates and sandstones. Some of these rocks may preserve fossil remains, especially sandstones and shales.
However, most urban fossils are preserved into limestone, a kind of sedimentary rock that is made mainly of calcic carbonate. Calcic carbonate is also the main chemical component of most shells and skeletons that are likely to fissilize. In fact, the main component of most limestones is a carbonate calcic from an organic origin.
In this section of fossiliferous construction stones you can find a list of sedimentary rocks used as construction materials that have a relevant fossiliferous content.

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Examples of sedimentary rocks: [A] sandstone, a siliciclastic rock, [B] close-up of a siliciclastic rock, [C] fossil-bearing limestone and [D] limestone used in a lampost.

Tips to improve visibility of urban fossils

Many urban fossils are more clear when the rock surface is wet. A thin layer of water increases color contrast and improves the perimeter of fossils against the rock, as well as their internal structure. This effect is particularly strong when the rock surface has been polished. On the contrary, water usually decreases the visibility of fossils on rough surfaces.
If you want to study urban fossils in detail and take high quality photographs, it is worth take with you a water spray to easily wet rock surfaces. Very often water reveals hidden fossils that are virtually undetectable when the rock surface is dry.

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Orthoceras fossil in a dry and wet surface (left and right)