Seven Slash Display

Python 3, 189 183 174 bytes

s="a%sa"%input()
while s[1:]:b,a,c,d,e,f,g=[c*(ord(n)>>int(s[~(n>"Ͱ")],16)&1)or" "for c,n in zip("\/"*4,"ΟϭŅͭͱͼϻ")];S=len(s)*"  ";print(S+a+b,c+d+"\n"+S+e+f+g);*s,_=s

The compression looks okay to me, but I'm having trouble coming up with a good way of ditching the seven variables...

Thankfully the spec is fairly relaxed on whitespace rules, because there's a lot of leading/trailing whitespace.

Expanded:

s="a%sa"%input()
while s[1:]:
  b,a,c,d,e,f,g=[c*(ord(n)>>int(s[~(n>"Ͱ")],16)&1)or" "
                 for c,n in zip("\/"*4,"ΟϭŅͭͱͼϻ")]
  S=len(s)*"  "
  print(S+a+b,c+d+"\n"+S+e+f+g)
  *s,_=s

Explanation

The segment positions represented by the variables are:

    ab               /\
    efg               /\
  ab cd            /\  /
  efg              \ \
ab cd            /\ \/
efg               /
 cd               \/

Each segment is encoded by a single 2-byte Unicode character. For example, ϻ encodes g's segment like so:

bin(ord("ϻ")) = bin(1019) = "0b1111111011"
                               ^^^^^^^^^^
                               9876543210

Indeed, 2 is the only digit to not use the bottom-right segment of a seven-segment display.

C, 1098 345 323 319 bytes

First Second Third attempt. Finally decided to ditch the screen buffer to save a few bytes. This program takes a parameter of digits and prints the digits in 7-segment format.

First time participant. Just for fun. Be gentle.

a[]={100489,2056,98569,67849,2440,67969,100737,2057,100745,67977},i,j,k,n,m;char*c=" /\\";
#define f(q,r) for(q=0;q<(r);q++)
#define P(w) putchar(w)
#define Q(d,i,j) P(c[a[v[1][d]-48]>>(i*3+j)*2&3])
main(w,char**v){f(i,n=strlen(v[1]))f(k,(m=n-i-1)?2:3){f(j,m*2)P(32);f(w,3)Q(m,k,w);if(!k&&i)f(w,2)Q(m+1,2,w+1);P(10);}}

Expanded, warning free:

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

int a[]={100489,2056,98569,67849,2440,67969,100737,2057,100745,67977};
char *c=" /\\";
#define f(q,r) for(q=0;q<(r);q++)
#define P(w) putchar(w)
#define Q(d,i,j) P(c[a[v[1][d]-48]>>(i*3+j)*2&3])
int main(int w, char **v)
{
    int i,j,k,n,m;
    f(i,n=strlen(v[1])) {
        m=n-i-1;
        f(k,m?2:3) {
            f(j,m*2) P(32);
            f(w,3) Q(m,k,w);
            if (!k&&i) f(w,2) Q(m+1,2,w+1);
            P(10);
        }
    }
}

JavaScript, 192 178 167 162 bytes

f=x=>{n=b="\n";for(k in x)for(i=0;i<8;)b+=("î\xA0Öô¸|~àþü".charCodeAt(x[k])>>i++&1?i%2?"/":"\\":" ")+(i%3?"":n+"  ".repeat(k));return b.split(n).reverse().join(n)}

Usage: f("1337"); will return

      /\
        \
    /\   
     /\
  /\  /
   /\
 \  /
  \
   

It uses features of ES6 and may have some implementation dependent behavior due to omission of semicolons and parentheses and such, but it works in Firefox.

Expanded:

f=x=>
{
    n = b = "\n";

    for (k in x)
        for (i=0; i<8;)
            b += ("î\xA0Öô¸|~àþü".charCodeAt(x[k]) >> i++ & 1? i%2? "/" : "\\" : " ") + (i%3? "" : n+"  ".repeat(k));

    return b.split(n).reverse().join(n)
}

Explanation:

l is an array containing 10 single byte characters which correspond to the shape of each digit. For example, the digit 0 is represented by the character î:

/\        11
\ \  -->  101  --> 11 101 110 = î
 \/       011

The input characters are used as keys to the array holding their shape representing counterparts, which get read bit by bit.