@links1=~s/.*/href\s*\=\s*\".*?\"/is;
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That wouldn't make much sense, because a character class can match any of its characters, but how should it be interpreted in the right side of s///? Should that mean all the characters it represents? Or a random one?
The following code does what you want:

for (@links1) {
  s/<\s*a\s.*?\b(href\s*=\s*".*?").*?>/<a $1>/i;
}
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Now let me explain your errors:

@links1=~   # that would work like scalar(@array)=~
s/ .*       # replace complete string with
 / href     # 'href'
   \s*      # as many spacey chars as possible
   \=       # no need to escape '='
   \s*      # as many spacey chars as possible
   \"       # no need to escape '"'
   .*?      # as few as possible (=no) anythings
   \"       # no need to escape '"'
 /isx;      # 's' is not needed, I added 'x'
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^~Django
"Why don't we ever challenge the spherical earth theory?"

I'm a beginner in Perl, but I think your breakdown of my code explains exactly what I want it to do. I went a little crazy with the escapes though. I'll need the s modifer for some other cases, though it's unnecessary for this example. I'd like the special meanings of the escaped characters on the right of the s/// to be used. I want the element in which the substitution is taking place (@links1[whatever]) to be referred to when Perl is assiging real text to the special characters. Not that you're solution is worse than my proposed one, but I was suprised to learn that mine didn't work.