Matrix rigidity was introduced by Valiant in 1977:
The rigidity $Rig_M(r)$ of boolean matrix $M$ over GF(2) is the smallest number of entries of $M$ that must be changed in order to reduce its rank over GF(2) down to $r$.
$Rig_M(r)$ has a deep connection to boolean function complexity & circuit lower bounds by a result of Razborov 1989 see eg . It also has a connection to finding locally correctable codes eg see 
It appears to me there is little to no published empirical analysis of $Rig_M(r)$ & that it could give some useful insight into its properties (thinking for example of the transition point research with SAT).
As a start I was interested in analyzing $Rig_M(r)$ using a simple greedy algorithm. The idea is as follows.
generate random matrixes $M$.
Compute Delta_row(i,j) where i,j are row (vectors) of $M$ and Delta_row(i,j) is the Hamming distance between the two rows
likewise calculate the Delta_col(i,j) where i,j are column vectors of $M$ (equivalent to calculating Delta_row(i,j) of the transpose of M).
next sort Delta_row(i,j) and Delta_col(i,j) and remove the column or row from $M$ with the "lowest value". (the lowest value is associated with a row or column pair, remove either row or column of the pair).
This greedy algorithm can be used to estimate $Rig_M(r)$ by giving a (fairly tight?) upper bound. One repeatedly reapplies the greedy algorithm until the matrix $M$ has been reduced to size $r$ and counts the sum of the Delta_row/col values of removed rows. and note that upon removing rows or columns, the new Delta_row and Delta_col arrays can be computed efficiently without recalculating the whole arrays.
how far off can the greedy algorithm be from the optimal value $Rig_M(r)$?
what kind of matrices maximize the difference between the greedy algorithm and the optimal rigidity measure?
can a modified local greedy algorithm find the optimal value $Rig_M(r)$